| L(s) = 1 | + 2-s − 3.07·3-s + 4-s + 0.116·5-s − 3.07·6-s − 1.89·7-s + 8-s + 6.47·9-s + 0.116·10-s − 0.612·11-s − 3.07·12-s + 0.224·13-s − 1.89·14-s − 0.358·15-s + 16-s + 5.05·17-s + 6.47·18-s − 3.64·19-s + 0.116·20-s + 5.84·21-s − 0.612·22-s − 23-s − 3.07·24-s − 4.98·25-s + 0.224·26-s − 10.7·27-s − 1.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.77·3-s + 0.5·4-s + 0.0520·5-s − 1.25·6-s − 0.718·7-s + 0.353·8-s + 2.15·9-s + 0.0367·10-s − 0.184·11-s − 0.888·12-s + 0.0622·13-s − 0.507·14-s − 0.0924·15-s + 0.250·16-s + 1.22·17-s + 1.52·18-s − 0.836·19-s + 0.0260·20-s + 1.27·21-s − 0.130·22-s − 0.208·23-s − 0.628·24-s − 0.997·25-s + 0.0440·26-s − 2.06·27-s − 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 - 0.116T + 5T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 + 0.612T + 11T^{2} \) |
| 13 | \( 1 - 0.224T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 3.64T + 19T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 3.49T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 5.00T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 4.68T + 61T^{2} \) |
| 67 | \( 1 - 7.56T + 67T^{2} \) |
| 71 | \( 1 + 4.99T + 71T^{2} \) |
| 73 | \( 1 - 3.92T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 6.02T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−9.777177320533948743327391102536, −8.163842822554660986628212867761, −7.19237572247964908533934297131, −6.37535282713304461768829179468, −5.90988305948188354344744722101, −5.13328441032050776187666534257, −4.28441384042129303332878983327, −3.24296646768537247413480074849, −1.59144127052718019034922804271, 0,
1.59144127052718019034922804271, 3.24296646768537247413480074849, 4.28441384042129303332878983327, 5.13328441032050776187666534257, 5.90988305948188354344744722101, 6.37535282713304461768829179468, 7.19237572247964908533934297131, 8.163842822554660986628212867761, 9.777177320533948743327391102536
