| L(s) = 1 | + 2-s + 0.124·3-s + 4-s − 2.23·5-s + 0.124·6-s + 3.94·7-s + 8-s − 2.98·9-s − 2.23·10-s − 0.760·11-s + 0.124·12-s − 3.14·13-s + 3.94·14-s − 0.278·15-s + 16-s − 1.62·17-s − 2.98·18-s − 19-s − 2.23·20-s + 0.491·21-s − 0.760·22-s + 8.96·23-s + 0.124·24-s − 0.00204·25-s − 3.14·26-s − 0.745·27-s + 3.94·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0719·3-s + 0.5·4-s − 0.999·5-s + 0.0508·6-s + 1.49·7-s + 0.353·8-s − 0.994·9-s − 0.706·10-s − 0.229·11-s + 0.0359·12-s − 0.872·13-s + 1.05·14-s − 0.0719·15-s + 0.250·16-s − 0.395·17-s − 0.703·18-s − 0.229·19-s − 0.499·20-s + 0.107·21-s − 0.162·22-s + 1.86·23-s + 0.0254·24-s − 0.000409·25-s − 0.617·26-s − 0.143·27-s + 0.745·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.124T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 + 0.760T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 6.31T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 9.13T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 + 1.98T + 83T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 - 0.0344T + 97T^{2} \) |
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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
−7.48070725378159420597086849293, −6.96799074612314041700364103818, −5.91014624471456126639443978402, −5.13548979770837234984873213411, −4.75623549847720324973292550318, −4.05435057002367909387104154114, −3.09323659111720358801747492746, −2.46776089818368532928841891479, −1.40875434633068266886272672975, 0,
1.40875434633068266886272672975, 2.46776089818368532928841891479, 3.09323659111720358801747492746, 4.05435057002367909387104154114, 4.75623549847720324973292550318, 5.13548979770837234984873213411, 5.91014624471456126639443978402, 6.96799074612314041700364103818, 7.48070725378159420597086849293
