| L(s) = 1 | − 2.62·2-s + 0.935·3-s + 4.90·4-s − 2.36·5-s − 2.45·6-s + 0.343·7-s − 7.64·8-s − 2.12·9-s + 6.21·10-s + 4.42·11-s + 4.58·12-s − 2.43·13-s − 0.903·14-s − 2.21·15-s + 10.2·16-s + 5.05·17-s + 5.58·18-s − 1.41·19-s − 11.5·20-s + 0.321·21-s − 11.6·22-s + 1.96·23-s − 7.14·24-s + 0.585·25-s + 6.40·26-s − 4.79·27-s + 1.68·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.539·3-s + 2.45·4-s − 1.05·5-s − 1.00·6-s + 0.129·7-s − 2.70·8-s − 0.708·9-s + 1.96·10-s + 1.33·11-s + 1.32·12-s − 0.675·13-s − 0.241·14-s − 0.570·15-s + 2.56·16-s + 1.22·17-s + 1.31·18-s − 0.324·19-s − 2.59·20-s + 0.0701·21-s − 2.48·22-s + 0.409·23-s − 1.45·24-s + 0.117·25-s + 1.25·26-s − 0.922·27-s + 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 0.935T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 - 0.343T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 31 | \( 1 - 0.410T + 31T^{2} \) |
| 37 | \( 1 + 1.20T + 37T^{2} \) |
| 41 | \( 1 + 8.28T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 8.43T + 59T^{2} \) |
| 61 | \( 1 + 4.77T + 61T^{2} \) |
| 67 | \( 1 + 3.01T + 67T^{2} \) |
| 71 | \( 1 - 8.62T + 71T^{2} \) |
| 73 | \( 1 - 1.28T + 73T^{2} \) |
| 79 | \( 1 - 1.49T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 2.47T + 97T^{2} \) |
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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
−8.207618456443228046126900941264, −7.66524667726498539968634364958, −7.01076235832735585525500253925, −6.31870303825831360810343582563, −5.22931449762692055211449108210, −3.83842533399637347653943289638, −3.20646543614116045786723034985, −2.19026901232322368454984135696, −1.15321172579857339528896298780, 0,
1.15321172579857339528896298780, 2.19026901232322368454984135696, 3.20646543614116045786723034985, 3.83842533399637347653943289638, 5.22931449762692055211449108210, 6.31870303825831360810343582563, 7.01076235832735585525500253925, 7.66524667726498539968634364958, 8.207618456443228046126900941264
