| L(s) = 1 | − 2-s + 4-s + 3·5-s − 3·7-s − 8-s − 3·10-s + 2·11-s − 5·13-s + 3·14-s + 16-s − 6·17-s + 3·20-s − 2·22-s + 2·23-s + 4·25-s + 5·26-s − 3·28-s − 6·29-s − 10·31-s − 32-s + 6·34-s − 9·35-s − 10·37-s − 3·40-s − 2·41-s + 4·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.13·7-s − 0.353·8-s − 0.948·10-s + 0.603·11-s − 1.38·13-s + 0.801·14-s + 1/4·16-s − 1.45·17-s + 0.670·20-s − 0.426·22-s + 0.417·23-s + 4/5·25-s + 0.980·26-s − 0.566·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s + 1.02·34-s − 1.52·35-s − 1.64·37-s − 0.474·40-s − 0.312·41-s + 0.609·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−9.174675330538307822169019064372, −8.820853343801771743435698779760, −7.17450713167885343099030707177, −6.93717508441912222809183241428, −5.95900466730196271449972715732, −5.22641373981494590954939993299, −3.80933035482911494201207656757, −2.56149928245518451776775111666, −1.83580907328715839320391152738, 0,
1.83580907328715839320391152738, 2.56149928245518451776775111666, 3.80933035482911494201207656757, 5.22641373981494590954939993299, 5.95900466730196271449972715732, 6.93717508441912222809183241428, 7.17450713167885343099030707177, 8.820853343801771743435698779760, 9.174675330538307822169019064372
