L(s) = 1 | + 2.34·2-s + 3-s + 3.48·4-s + 2.90·5-s + 2.34·6-s + 3.05·7-s + 3.47·8-s + 9-s + 6.81·10-s − 0.932·11-s + 3.48·12-s − 13-s + 7.15·14-s + 2.90·15-s + 1.17·16-s + 0.0973·17-s + 2.34·18-s − 1.46·19-s + 10.1·20-s + 3.05·21-s − 2.18·22-s − 9.37·23-s + 3.47·24-s + 3.46·25-s − 2.34·26-s + 27-s + 10.6·28-s + ⋯ |
L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.74·4-s + 1.30·5-s + 0.956·6-s + 1.15·7-s + 1.22·8-s + 0.333·9-s + 2.15·10-s − 0.281·11-s + 1.00·12-s − 0.277·13-s + 1.91·14-s + 0.751·15-s + 0.293·16-s + 0.0236·17-s + 0.552·18-s − 0.336·19-s + 2.26·20-s + 0.666·21-s − 0.465·22-s − 1.95·23-s + 0.709·24-s + 0.692·25-s − 0.459·26-s + 0.192·27-s + 2.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.675812191\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.675812191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 0.932T + 11T^{2} \) |
| 17 | \( 1 - 0.0973T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 + 9.37T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 8.55T + 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 + 0.0979T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 0.751T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 + 4.82T + 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.181299051846997791395350760262, −7.73856268590794454523539158853, −6.52353512288990729508107724156, −6.13403179308308474362380567879, −5.31172382107860726322027389891, −4.66848658350278013958469031417, −4.08028061460631799681530746947, −2.89796802638322471386631128370, −2.25455206569406209030034115963, −1.59792272943481909116189159305,
1.59792272943481909116189159305, 2.25455206569406209030034115963, 2.89796802638322471386631128370, 4.08028061460631799681530746947, 4.66848658350278013958469031417, 5.31172382107860726322027389891, 6.13403179308308474362380567879, 6.52353512288990729508107724156, 7.73856268590794454523539158853, 8.181299051846997791395350760262