L(s) = 1 | + 2-s + 2.87·3-s + 4-s − 4.41·5-s + 2.87·6-s − 0.694·7-s + 8-s + 5.29·9-s − 4.41·10-s − 5.59·11-s + 2.87·12-s − 0.162·13-s − 0.694·14-s − 12.7·15-s + 16-s + 3.10·17-s + 5.29·18-s + 3.38·19-s − 4.41·20-s − 2·21-s − 5.59·22-s + 3.24·23-s + 2.87·24-s + 14.4·25-s − 0.162·26-s + 6.59·27-s − 0.694·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.66·3-s + 0.5·4-s − 1.97·5-s + 1.17·6-s − 0.262·7-s + 0.353·8-s + 1.76·9-s − 1.39·10-s − 1.68·11-s + 0.831·12-s − 0.0450·13-s − 0.185·14-s − 3.27·15-s + 0.250·16-s + 0.753·17-s + 1.24·18-s + 0.777·19-s − 0.986·20-s − 0.436·21-s − 1.19·22-s + 0.677·23-s + 0.587·24-s + 2.89·25-s − 0.0318·26-s + 1.26·27-s − 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.855192648\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.855192648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 + 4.41T + 5T^{2} \) |
| 7 | \( 1 + 0.694T + 7T^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 + 0.162T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 0.241T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 41 | \( 1 + 0.694T + 41T^{2} \) |
| 43 | \( 1 + 6.86T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 + 4.18T + 53T^{2} \) |
| 59 | \( 1 - 0.694T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 - 8.08T + 83T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−13.22488590432464789396231546973, −12.56253973860395035539764903199, −11.41234339634995860282874014626, −10.19833252033824738945625497915, −8.685743901035079225314272845138, −7.73128921744132614398563094789, −7.35884855139902712658880863108, −4.92966724000788216658445541653, −3.59596559245939116333038332560, −2.93232000032698795456793095831,
2.93232000032698795456793095831, 3.59596559245939116333038332560, 4.92966724000788216658445541653, 7.35884855139902712658880863108, 7.73128921744132614398563094789, 8.685743901035079225314272845138, 10.19833252033824738945625497915, 11.41234339634995860282874014626, 12.56253973860395035539764903199, 13.22488590432464789396231546973