| L(s) = 1 | − 2-s + 2.75·3-s + 4-s + 1.27·5-s − 2.75·6-s + 0.521·7-s − 8-s + 4.61·9-s − 1.27·10-s − 2.91·11-s + 2.75·12-s − 1.67·13-s − 0.521·14-s + 3.52·15-s + 16-s − 1.41·17-s − 4.61·18-s − 19-s + 1.27·20-s + 1.43·21-s + 2.91·22-s − 2.74·23-s − 2.75·24-s − 3.37·25-s + 1.67·26-s + 4.45·27-s + 0.521·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.59·3-s + 0.5·4-s + 0.570·5-s − 1.12·6-s + 0.197·7-s − 0.353·8-s + 1.53·9-s − 0.403·10-s − 0.879·11-s + 0.796·12-s − 0.464·13-s − 0.139·14-s + 0.908·15-s + 0.250·16-s − 0.342·17-s − 1.08·18-s − 0.229·19-s + 0.285·20-s + 0.313·21-s + 0.622·22-s − 0.573·23-s − 0.563·24-s − 0.674·25-s + 0.328·26-s + 0.857·27-s + 0.0985·28-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 | \( 1 - 2.75T + 3T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 0.521T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 7.50T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 - 6.88T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 + 0.605T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 1.50T + 67T^{2} \) |
| 71 | \( 1 - 5.84T + 71T^{2} \) |
| 73 | \( 1 + 5.63T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 + 1.38T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 7.33T + 97T^{2} \) |
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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
−7.59832358332294968434194119520, −7.30138445663817772100118166407, −6.19528248190756385565869076194, −5.50696996100411837558239109053, −4.51813962030923096642929483454, −3.63413864099600864677280597721, −2.86789285801688233326948374340, −2.09208298905948603940566301683, −1.72588905494489394509277718328, 0,
1.72588905494489394509277718328, 2.09208298905948603940566301683, 2.86789285801688233326948374340, 3.63413864099600864677280597721, 4.51813962030923096642929483454, 5.50696996100411837558239109053, 6.19528248190756385565869076194, 7.30138445663817772100118166407, 7.59832358332294968434194119520
