| L(s) = 1 | + 3·3-s − 5·5-s + 4·7-s + 9·9-s + 6·11-s + 44·13-s − 15·15-s − 84·17-s + 8·19-s + 12·21-s − 48·23-s + 25·25-s + 27·27-s − 90·29-s − 166·31-s + 18·33-s − 20·35-s − 156·37-s + 132·39-s + 166·41-s − 460·43-s − 45·45-s + 448·47-s − 327·49-s − 252·51-s + 470·53-s − 30·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.215·7-s + 1/3·9-s + 0.164·11-s + 0.938·13-s − 0.258·15-s − 1.19·17-s + 0.0965·19-s + 0.124·21-s − 0.435·23-s + 1/5·25-s + 0.192·27-s − 0.576·29-s − 0.961·31-s + 0.0949·33-s − 0.0965·35-s − 0.693·37-s + 0.541·39-s + 0.632·41-s − 1.63·43-s − 0.149·45-s + 1.39·47-s − 0.953·49-s − 0.691·51-s + 1.21·53-s − 0.0735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 8 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 166 T + p^{3} T^{2} \) |
| 37 | \( 1 + 156 T + p^{3} T^{2} \) |
| 41 | \( 1 - 166 T + p^{3} T^{2} \) |
| 43 | \( 1 + 460 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 - 470 T + p^{3} T^{2} \) |
| 59 | \( 1 - 26 T + p^{3} T^{2} \) |
| 61 | \( 1 - 206 T + p^{3} T^{2} \) |
| 67 | \( 1 + 548 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 30 T + p^{3} T^{2} \) |
| 79 | \( 1 + 750 T + p^{3} T^{2} \) |
| 83 | \( 1 + 4 T + p^{3} T^{2} \) |
| 89 | \( 1 + 186 T + p^{3} T^{2} \) |
| 97 | \( 1 - 530 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.643423497368832829665913917168, −7.71379578285398550783329568990, −7.00103966709819475863458978856, −6.15714796241499391932882928826, −5.12826373465534996579168197212, −4.10376920951538907644697368025, −3.55835073233289306493657518613, −2.37005341192056672670135619088, −1.40091551145768825362113310234, 0,
1.40091551145768825362113310234, 2.37005341192056672670135619088, 3.55835073233289306493657518613, 4.10376920951538907644697368025, 5.12826373465534996579168197212, 6.15714796241499391932882928826, 7.00103966709819475863458978856, 7.71379578285398550783329568990, 8.643423497368832829665913917168
