L(s) = 1 | − 2-s + 3.32·3-s + 4-s − 1.53·5-s − 3.32·6-s − 7-s − 8-s + 8.07·9-s + 1.53·10-s − 0.220·11-s + 3.32·12-s − 0.413·13-s + 14-s − 5.11·15-s + 16-s − 4.27·17-s − 8.07·18-s + 0.560·19-s − 1.53·20-s − 3.32·21-s + 0.220·22-s + 6.61·23-s − 3.32·24-s − 2.63·25-s + 0.413·26-s + 16.8·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.92·3-s + 0.5·4-s − 0.687·5-s − 1.35·6-s − 0.377·7-s − 0.353·8-s + 2.69·9-s + 0.486·10-s − 0.0664·11-s + 0.960·12-s − 0.114·13-s + 0.267·14-s − 1.32·15-s + 0.250·16-s − 1.03·17-s − 1.90·18-s + 0.128·19-s − 0.343·20-s − 0.726·21-s + 0.0469·22-s + 1.37·23-s − 0.679·24-s − 0.527·25-s + 0.0810·26-s + 3.24·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.572439104\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.572439104\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 3 | \( 1 - 3.32T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 11 | \( 1 + 0.220T + 11T^{2} \) |
| 13 | \( 1 + 0.413T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 19 | \( 1 - 0.560T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 - 6.99T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 + 0.779T + 43T^{2} \) |
| 47 | \( 1 - 5.40T + 47T^{2} \) |
| 53 | \( 1 - 8.45T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 - 8.00T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 1.20T + 73T^{2} \) |
| 79 | \( 1 - 0.0748T + 79T^{2} \) |
| 83 | \( 1 + 6.38T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.253656939543277148078167878935, −7.44502655629872052468616451286, −7.15557339693177507111142662018, −6.35147574279672401181356727036, −4.97251568433581729101809147547, −4.06745092014687789964121195729, −3.49491998937107372579874796118, −2.66690130212002233136163626244, −2.08217549301289261417115868223, −0.852565163474016829081619831074,
0.852565163474016829081619831074, 2.08217549301289261417115868223, 2.66690130212002233136163626244, 3.49491998937107372579874796118, 4.06745092014687789964121195729, 4.97251568433581729101809147547, 6.35147574279672401181356727036, 7.15557339693177507111142662018, 7.44502655629872052468616451286, 8.253656939543277148078167878935