| L(s) = 1 | − 5·3-s + 50·5-s − 98·7-s − 185·9-s − 415·11-s + 429·13-s − 250·15-s + 1.31e3·17-s − 1.91e3·19-s + 490·21-s + 1.33e3·23-s + 1.87e3·25-s + 760·27-s − 1.13e3·29-s + 5.47e3·31-s + 2.07e3·33-s − 4.90e3·35-s − 9.15e3·37-s − 2.14e3·39-s − 7.82e3·41-s − 2.07e3·43-s − 9.25e3·45-s + 8.50e3·47-s + 7.20e3·49-s − 6.59e3·51-s − 3.42e4·53-s − 2.07e4·55-s + ⋯ |
| L(s) = 1 | − 0.320·3-s + 0.894·5-s − 0.755·7-s − 0.761·9-s − 1.03·11-s + 0.704·13-s − 0.286·15-s + 1.10·17-s − 1.21·19-s + 0.242·21-s + 0.525·23-s + 3/5·25-s + 0.200·27-s − 0.249·29-s + 1.02·31-s + 0.331·33-s − 0.676·35-s − 1.09·37-s − 0.225·39-s − 0.726·41-s − 0.171·43-s − 0.680·45-s + 0.561·47-s + 3/7·49-s − 0.355·51-s − 1.67·53-s − 0.924·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 70 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 415 T + 317458 T^{2} + 415 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 33 p T + 754444 T^{2} - 33 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1319 T + 2082148 T^{2} - 1319 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1918 T + 2203758 T^{2} + 1918 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 58 p T + 10606846 T^{2} - 58 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 39 p T + 38900908 T^{2} + 39 p^{6} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5472 T + 8095294 T^{2} - 5472 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9156 T + 152420398 T^{2} + 9156 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7822 T + 243078274 T^{2} + 7822 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2078 T - 70429762 T^{2} + 2078 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 181 p T - 114195314 T^{2} - 181 p^{6} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 34242 T + 846139498 T^{2} + 34242 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6952 T + 1381163878 T^{2} + 6952 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 45138 T + 2092145242 T^{2} + 45138 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 54556 T + 3349390598 T^{2} - 54556 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 62272 T + 3502450894 T^{2} - 62272 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 40896 T + 3740866126 T^{2} - 40896 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 62069 T + 5500962758 T^{2} + 62069 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 189600 T + 16864958710 T^{2} + 189600 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 64190 T + 10032531898 T^{2} - 64190 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 155547 T + 22978754524 T^{2} - 155547 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.662863099818797725432573087190, −9.571693969788556555142301050876, −8.837129944107051632967414787126, −8.370895437128493041019006777810, −8.206356728584180509694023673956, −7.52593870498822479765060161590, −6.81820668569644275176404756417, −6.59203842449609745957997425523, −5.89621882924403062431394195558, −5.83045269150947931037442139669, −5.13716378062526209022329426172, −4.86359321744990403780691768581, −3.94688784427992563561491519719, −3.42982930190889232764821188990, −2.77288362971831148770253333890, −2.53705547032818661024207923637, −1.60199308081565307535669065697, −1.10968522333673518096152742048, 0, 0,
1.10968522333673518096152742048, 1.60199308081565307535669065697, 2.53705547032818661024207923637, 2.77288362971831148770253333890, 3.42982930190889232764821188990, 3.94688784427992563561491519719, 4.86359321744990403780691768581, 5.13716378062526209022329426172, 5.83045269150947931037442139669, 5.89621882924403062431394195558, 6.59203842449609745957997425523, 6.81820668569644275176404756417, 7.52593870498822479765060161590, 8.206356728584180509694023673956, 8.370895437128493041019006777810, 8.837129944107051632967414787126, 9.571693969788556555142301050876, 9.662863099818797725432573087190
