| L(s) = 1 | − 2·5-s − 4·7-s − 2·11-s + 4·17-s − 6·19-s − 2·23-s + 8·29-s − 8·31-s + 8·35-s − 2·37-s + 12·41-s − 6·43-s − 12·47-s − 2·49-s + 2·53-s + 4·55-s + 6·61-s + 14·67-s + 12·71-s − 4·73-s + 8·77-s + 4·79-s + 22·83-s − 8·85-s + 12·95-s + 8·97-s + 8·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.603·11-s + 0.970·17-s − 1.37·19-s − 0.417·23-s + 1.48·29-s − 1.43·31-s + 1.35·35-s − 0.328·37-s + 1.87·41-s − 0.914·43-s − 1.75·47-s − 2/7·49-s + 0.274·53-s + 0.539·55-s + 0.768·61-s + 1.71·67-s + 1.42·71-s − 0.468·73-s + 0.911·77-s + 0.450·79-s + 2.41·83-s − 0.867·85-s + 1.23·95-s + 0.812·97-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10969344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10969344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7709070896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7709070896\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 100 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 120 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 280 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 182 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.644521081203940132928920917257, −8.424069812781998567754931990408, −8.078033923840736098921709271125, −7.73676029334730807771082332480, −7.41154179483305019575461866462, −6.87244969850281025297828776909, −6.57221249378169992587845957790, −6.26334574240706933803283295482, −5.95512543701163605323972317764, −5.40150110193823315521106959512, −4.84564180330569330362921038899, −4.73954111013446456761199441200, −3.91824892151757401687847009011, −3.73180153015370892915439964831, −3.28759791368897609155983525283, −3.03907403530293977978765569322, −2.19909847541991032294005955188, −2.03313322408017026517600784716, −0.900766288538204409050771170667, −0.33317704445890312450779789993,
0.33317704445890312450779789993, 0.900766288538204409050771170667, 2.03313322408017026517600784716, 2.19909847541991032294005955188, 3.03907403530293977978765569322, 3.28759791368897609155983525283, 3.73180153015370892915439964831, 3.91824892151757401687847009011, 4.73954111013446456761199441200, 4.84564180330569330362921038899, 5.40150110193823315521106959512, 5.95512543701163605323972317764, 6.26334574240706933803283295482, 6.57221249378169992587845957790, 6.87244969850281025297828776909, 7.41154179483305019575461866462, 7.73676029334730807771082332480, 8.078033923840736098921709271125, 8.424069812781998567754931990408, 8.644521081203940132928920917257
