| L(s) = 1 | − 3-s − 2·9-s − 5·11-s + 3·13-s − 5·17-s + 2·19-s − 2·23-s + 5·27-s − 3·29-s + 6·31-s + 5·33-s − 2·37-s − 3·39-s − 2·41-s − 6·43-s − 3·47-s + 5·51-s + 12·53-s − 2·57-s − 12·59-s − 6·61-s + 12·67-s + 2·69-s + 8·71-s − 6·73-s + 3·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.50·11-s + 0.832·13-s − 1.21·17-s + 0.458·19-s − 0.417·23-s + 0.962·27-s − 0.557·29-s + 1.07·31-s + 0.870·33-s − 0.328·37-s − 0.480·39-s − 0.312·41-s − 0.914·43-s − 0.437·47-s + 0.700·51-s + 1.64·53-s − 0.264·57-s − 1.56·59-s − 0.768·61-s + 1.46·67-s + 0.240·69-s + 0.949·71-s − 0.702·73-s + 0.337·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.08149297487694, −13.63834433180345, −13.39144853598936, −12.77493331619070, −12.28532653810146, −11.55831486046357, −11.38656421851227, −10.79249388626930, −10.34698878320469, −9.953313337942448, −9.095269503843171, −8.691490477429624, −8.162284212917518, −7.740726531465208, −7.013404878720659, −6.395170752064685, −6.045919444608809, −5.321781843284925, −5.044857356929490, −4.373629560309244, −3.610247822141523, −2.965251443422527, −2.416290845659329, −1.695368255078688, −0.6758963296560568, 0,
0.6758963296560568, 1.695368255078688, 2.416290845659329, 2.965251443422527, 3.610247822141523, 4.373629560309244, 5.044857356929490, 5.321781843284925, 6.045919444608809, 6.395170752064685, 7.013404878720659, 7.740726531465208, 8.162284212917518, 8.691490477429624, 9.095269503843171, 9.953313337942448, 10.34698878320469, 10.79249388626930, 11.38656421851227, 11.55831486046357, 12.28532653810146, 12.77493331619070, 13.39144853598936, 13.63834433180345, 14.08149297487694
