| L(s) = 1 | + 1.34·2-s − 0.194·4-s + 4.16·5-s + 3.85·7-s − 2.94·8-s + 5.60·10-s + 1.46·11-s + 5.38·13-s + 5.17·14-s − 3.57·16-s + 4.09·17-s − 19-s − 0.810·20-s + 1.96·22-s − 2.83·23-s + 12.3·25-s + 7.23·26-s − 0.748·28-s + 3.48·29-s + 6.28·31-s + 1.09·32-s + 5.49·34-s + 16.0·35-s − 7.07·37-s − 1.34·38-s − 12.2·40-s − 1.57·41-s + ⋯ |
| L(s) = 1 | + 0.950·2-s − 0.0972·4-s + 1.86·5-s + 1.45·7-s − 1.04·8-s + 1.77·10-s + 0.441·11-s + 1.49·13-s + 1.38·14-s − 0.893·16-s + 0.992·17-s − 0.229·19-s − 0.181·20-s + 0.419·22-s − 0.591·23-s + 2.47·25-s + 1.41·26-s − 0.141·28-s + 0.647·29-s + 1.12·31-s + 0.193·32-s + 0.942·34-s + 2.71·35-s − 1.16·37-s − 0.217·38-s − 1.94·40-s − 0.246·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.997721989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.997721989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 5.38T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 - 6.28T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 - 2.95T + 83T^{2} \) |
| 89 | \( 1 + 9.72T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
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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
−7.969532712159898102993758771310, −6.73180652024261003425894918821, −6.16237150415863055732470318285, −5.69476943405978747208152753245, −5.05512543766723869303479306441, −4.52286744780012440005073264628, −3.57857527808144784070935591729, −2.76195539218320149071405813250, −1.68675426180142752755829558110, −1.25159653554276672849803726648,
1.25159653554276672849803726648, 1.68675426180142752755829558110, 2.76195539218320149071405813250, 3.57857527808144784070935591729, 4.52286744780012440005073264628, 5.05512543766723869303479306441, 5.69476943405978747208152753245, 6.16237150415863055732470318285, 6.73180652024261003425894918821, 7.969532712159898102993758771310
