| L(s) = 1 | + 2-s + 3.35·3-s + 4-s − 5-s + 3.35·6-s + 0.572·7-s + 8-s + 8.23·9-s − 10-s + 11-s + 3.35·12-s + 3.12·13-s + 0.572·14-s − 3.35·15-s + 16-s − 1.93·17-s + 8.23·18-s + 2.54·19-s − 20-s + 1.91·21-s + 22-s − 8.34·23-s + 3.35·24-s + 25-s + 3.12·26-s + 17.5·27-s + 0.572·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.93·3-s + 0.5·4-s − 0.447·5-s + 1.36·6-s + 0.216·7-s + 0.353·8-s + 2.74·9-s − 0.316·10-s + 0.301·11-s + 0.967·12-s + 0.866·13-s + 0.153·14-s − 0.865·15-s + 0.250·16-s − 0.470·17-s + 1.94·18-s + 0.584·19-s − 0.223·20-s + 0.418·21-s + 0.213·22-s − 1.73·23-s + 0.684·24-s + 0.200·25-s + 0.612·26-s + 3.37·27-s + 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.497203516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.497203516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 3.35T + 3T^{2} \) |
| 7 | \( 1 - 0.572T + 7T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - 7.55T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 + 5.34T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 2.76T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 6.22T + 67T^{2} \) |
| 71 | \( 1 + 3.95T + 71T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 + 8.11T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−7.86234837701220757899257232366, −7.32420932423688499438367907418, −6.58406908931673034034744541369, −5.78061745152132130055952265923, −4.56761171761928556604880877945, −4.12446141185656848835746551734, −3.54305914138417366053390998811, −2.83350174065065001945976540545, −2.05836294805315847545914603626, −1.23288670423422825976193821476,
1.23288670423422825976193821476, 2.05836294805315847545914603626, 2.83350174065065001945976540545, 3.54305914138417366053390998811, 4.12446141185656848835746551734, 4.56761171761928556604880877945, 5.78061745152132130055952265923, 6.58406908931673034034744541369, 7.32420932423688499438367907418, 7.86234837701220757899257232366
