| L(s) = 1 | − 16·2-s + 64·4-s − 14·5-s − 1.84e3·7-s + 1.02e3·8-s + 224·10-s + 2.40e3·11-s + 2.14e4·13-s + 2.95e4·14-s − 1.63e4·16-s − 3.50e4·17-s + 2.40e3·19-s − 896·20-s − 3.85e4·22-s − 6.16e4·23-s − 2.97e4·25-s − 3.43e5·26-s − 1.18e5·28-s + 1.91e5·29-s + 1.66e5·31-s + 6.55e4·32-s + 5.61e5·34-s + 2.58e4·35-s + 5.59e5·37-s − 3.85e4·38-s − 1.43e4·40-s − 1.61e6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.0500·5-s − 2.03·7-s + 0.707·8-s + 0.0708·10-s + 0.545·11-s + 2.70·13-s + 2.87·14-s − 16-s − 1.73·17-s + 0.0805·19-s − 0.0250·20-s − 0.771·22-s − 1.05·23-s − 0.380·25-s − 3.82·26-s − 1.01·28-s + 1.45·29-s + 1.00·31-s + 0.353·32-s + 2.45·34-s + 0.101·35-s + 1.81·37-s − 0.113·38-s − 0.0354·40-s − 3.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.857316330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.857316330\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 264 p T + 5122 p^{3} T^{2} + 264 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 14 T + 29901 T^{2} - 520674 p T^{3} - 209457644 p^{2} T^{4} - 520674 p^{8} T^{5} + 29901 p^{14} T^{6} + 14 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2408 T + 10061967 T^{2} + 104116730760 T^{3} - 474568035257576 T^{4} + 104116730760 p^{7} T^{5} + 10061967 p^{14} T^{6} - 2408 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10724 T + 152410814 T^{2} - 10724 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 35098 T + 212929257 T^{2} + 6958634809098 T^{3} + 364377865570683988 T^{4} + 6958634809098 p^{7} T^{5} + 212929257 p^{14} T^{6} + 35098 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2408 T - 1156135049 T^{2} + 1506950395720 T^{3} + 545899719750428152 T^{4} + 1506950395720 p^{7} T^{5} - 1156135049 p^{14} T^{6} - 2408 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 61684 T - 2114477901 T^{2} - 54914621238708 T^{3} + 10491311795669237608 T^{4} - 54914621238708 p^{7} T^{5} - 2114477901 p^{14} T^{6} + 61684 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 95660 T + 34197541822 T^{2} - 95660 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 166012 T - 1067536483 p T^{2} - 934379765212740 T^{3} + \)\(21\!\cdots\!84\)\( T^{4} - 934379765212740 p^{7} T^{5} - 1067536483 p^{15} T^{6} - 166012 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 559814 T + 77524081237 T^{2} - 25753615570568702 T^{3} + \)\(16\!\cdots\!24\)\( T^{4} - 25753615570568702 p^{7} T^{5} + 77524081237 p^{14} T^{6} - 559814 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 805980 T + 544490570278 T^{2} + 805980 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 6232 p T + 534162600534 T^{2} - 6232 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1769292 T + 1340647882547 T^{2} - 1373855340249153972 T^{3} + \)\(13\!\cdots\!88\)\( T^{4} - 1373855340249153972 p^{7} T^{5} + 1340647882547 p^{14} T^{6} - 1769292 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2317194 T + 2139024918053 T^{2} - 2041310819323319346 T^{3} + \)\(27\!\cdots\!88\)\( T^{4} - 2041310819323319346 p^{7} T^{5} + 2139024918053 p^{14} T^{6} - 2317194 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 660352 T - 4531445555481 T^{2} + 6466596179869056 T^{3} + \)\(17\!\cdots\!88\)\( T^{4} + 6466596179869056 p^{7} T^{5} - 4531445555481 p^{14} T^{6} - 660352 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 1463042 T - 2995701633619 T^{2} - 1681462677906192678 T^{3} + \)\(97\!\cdots\!04\)\( T^{4} - 1681462677906192678 p^{7} T^{5} - 2995701633619 p^{14} T^{6} + 1463042 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1784280 T - 5420427629185 T^{2} + 6275920241430481080 T^{3} + \)\(18\!\cdots\!96\)\( T^{4} + 6275920241430481080 p^{7} T^{5} - 5420427629185 p^{14} T^{6} - 1784280 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 274400 T - 1862728669394 T^{2} + 274400 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4549062 T - 5976062559175 T^{2} + 20813007667450176150 T^{3} + \)\(36\!\cdots\!84\)\( T^{4} + 20813007667450176150 p^{7} T^{5} - 5976062559175 p^{14} T^{6} + 4549062 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8673964 T + 24597682454603 T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!04\)\( T^{4} - \)\(10\!\cdots\!00\)\( p^{7} T^{5} + 24597682454603 p^{14} T^{6} - 8673964 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 10594360 T + 68478299343430 T^{2} - 10594360 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1779750 T - 23603446896199 T^{2} + \)\(10\!\cdots\!50\)\( T^{3} - \)\(13\!\cdots\!40\)\( T^{4} + \)\(10\!\cdots\!50\)\( p^{7} T^{5} - 23603446896199 p^{14} T^{6} - 1779750 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 1748964 T + 72590049947446 T^{2} + 1748964 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.750317872947281017026755725205, −8.192889625861135033333754989084, −8.073396063702544279082094821950, −7.59488898765965087377625331655, −7.46674334760732718818711471659, −6.75709521933072660312673814392, −6.53604804164692489720078473348, −6.48513799198602969893874127221, −6.45881752273243261451555126526, −5.81945056687442256532136741687, −5.78987328109233567441657884884, −5.07663771282332683277123990818, −4.80416127382966785317087302823, −4.19370296260807928982347510408, −3.89170621444564827960455128885, −3.83848895228160037796387975676, −3.58863433550612854807905531198, −2.81487259951558639872125758452, −2.69522305804815922169809783816, −2.21945789622096764319913015185, −1.67430820373114680327928779499, −1.32664873855585998892525541528, −0.72849297592262513352046855995, −0.63038654752395917468142909362, −0.38942249018279724809480915530,
0.38942249018279724809480915530, 0.63038654752395917468142909362, 0.72849297592262513352046855995, 1.32664873855585998892525541528, 1.67430820373114680327928779499, 2.21945789622096764319913015185, 2.69522305804815922169809783816, 2.81487259951558639872125758452, 3.58863433550612854807905531198, 3.83848895228160037796387975676, 3.89170621444564827960455128885, 4.19370296260807928982347510408, 4.80416127382966785317087302823, 5.07663771282332683277123990818, 5.78987328109233567441657884884, 5.81945056687442256532136741687, 6.45881752273243261451555126526, 6.48513799198602969893874127221, 6.53604804164692489720078473348, 6.75709521933072660312673814392, 7.46674334760732718818711471659, 7.59488898765965087377625331655, 8.073396063702544279082094821950, 8.192889625861135033333754989084, 8.750317872947281017026755725205
