| L(s) = 1 | − 2-s − 3.06·3-s + 4-s − 3.85·5-s + 3.06·6-s + 4.92·7-s − 8-s + 6.39·9-s + 3.85·10-s − 3.45·11-s − 3.06·12-s − 6.10·13-s − 4.92·14-s + 11.8·15-s + 16-s + 6.53·17-s − 6.39·18-s + 8.49·19-s − 3.85·20-s − 15.1·21-s + 3.45·22-s − 3.16·23-s + 3.06·24-s + 9.83·25-s + 6.10·26-s − 10.4·27-s + 4.92·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.76·3-s + 0.5·4-s − 1.72·5-s + 1.25·6-s + 1.86·7-s − 0.353·8-s + 2.13·9-s + 1.21·10-s − 1.04·11-s − 0.884·12-s − 1.69·13-s − 1.31·14-s + 3.04·15-s + 0.250·16-s + 1.58·17-s − 1.50·18-s + 1.94·19-s − 0.861·20-s − 3.29·21-s + 0.735·22-s − 0.660·23-s + 0.625·24-s + 1.96·25-s + 1.19·26-s − 2.00·27-s + 0.931·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5515520477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5515520477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 3.06T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 + 6.10T + 13T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 19 | \( 1 - 8.49T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 - 9.85T + 43T^{2} \) |
| 47 | \( 1 + 0.756T + 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 - 7.49T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 - 8.05T + 67T^{2} \) |
| 71 | \( 1 + 2.20T + 71T^{2} \) |
| 73 | \( 1 - 0.461T + 73T^{2} \) |
| 79 | \( 1 - 2.70T + 79T^{2} \) |
| 83 | \( 1 + 9.44T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
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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
−7.85993052619829242475820903907, −7.37031755015694234447418227395, −7.18597615483672183795092708805, −5.69655688861317757477527468231, −5.07162274941617391880812612488, −4.88521062497687150578599381457, −3.87762018246292300234722972771, −2.63751127806829596858233709494, −1.22601656224034608118171742288, −0.56476201079609761768804107838,
0.56476201079609761768804107838, 1.22601656224034608118171742288, 2.63751127806829596858233709494, 3.87762018246292300234722972771, 4.88521062497687150578599381457, 5.07162274941617391880812612488, 5.69655688861317757477527468231, 7.18597615483672183795092708805, 7.37031755015694234447418227395, 7.85993052619829242475820903907
