| L(s) = 1 | + 2.07·2-s + 2.32·4-s − 0.602·7-s + 0.670·8-s + 3.75·11-s − 3.15·13-s − 1.25·14-s − 3.25·16-s − 5.57·17-s − 3.67·19-s + 7.80·22-s + 5.90·23-s − 6.54·26-s − 1.39·28-s + 29-s − 3.09·31-s − 8.09·32-s − 11.5·34-s − 9.93·37-s − 7.63·38-s + 0.0283·41-s + 5.05·43-s + 8.71·44-s + 12.2·46-s − 2.15·47-s − 6.63·49-s − 7.31·52-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.16·4-s − 0.227·7-s + 0.237·8-s + 1.13·11-s − 0.873·13-s − 0.334·14-s − 0.812·16-s − 1.35·17-s − 0.842·19-s + 1.66·22-s + 1.23·23-s − 1.28·26-s − 0.264·28-s + 0.185·29-s − 0.555·31-s − 1.43·32-s − 1.98·34-s − 1.63·37-s − 1.23·38-s + 0.00443·41-s + 0.770·43-s + 1.31·44-s + 1.81·46-s − 0.314·47-s − 0.948·49-s − 1.01·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 7 | \( 1 + 0.602T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 + 5.57T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 - 0.0283T + 41T^{2} \) |
| 43 | \( 1 - 5.05T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 9.99T + 59T^{2} \) |
| 61 | \( 1 + 5.22T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 - 9.33T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.20993138234823062778039892090, −6.69753410188799093729545654694, −6.30188263788654961924359227320, −5.30607853523701675203518630657, −4.72399464498383652840647426468, −4.12243707137649188093188200487, −3.37965066541536046710775592995, −2.56961685036824597309077214751, −1.70349115438103068567087627505, 0,
1.70349115438103068567087627505, 2.56961685036824597309077214751, 3.37965066541536046710775592995, 4.12243707137649188093188200487, 4.72399464498383652840647426468, 5.30607853523701675203518630657, 6.30188263788654961924359227320, 6.69753410188799093729545654694, 7.20993138234823062778039892090
