L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 3.46·7-s − 8-s + 9-s + 4.15·11-s − 12-s + 3.73·13-s − 3.46·14-s + 16-s − 4.63·17-s − 18-s + 5.88·19-s − 3.46·21-s − 4.15·22-s − 4.36·23-s + 24-s − 3.73·26-s − 27-s + 3.46·28-s + 8.03·29-s + 0.895·31-s − 32-s − 4.15·33-s + 4.63·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.30·7-s − 0.353·8-s + 0.333·9-s + 1.25·11-s − 0.288·12-s + 1.03·13-s − 0.926·14-s + 0.250·16-s − 1.12·17-s − 0.235·18-s + 1.34·19-s − 0.756·21-s − 0.885·22-s − 0.909·23-s + 0.204·24-s − 0.733·26-s − 0.192·27-s + 0.654·28-s + 1.49·29-s + 0.160·31-s − 0.176·32-s − 0.723·33-s + 0.794·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733499582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733499582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 4.63T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 - 0.895T + 31T^{2} \) |
| 41 | \( 1 + 2.89T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 7.19T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7.06T + 97T^{2} \) |
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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
−8.344879216440505112341823815918, −7.42831316722326391535076090152, −6.80320970906022861265370652393, −6.08710113366665458314298997586, −5.41180118180413728835689362746, −4.41283997663094533202593594826, −3.88961624172070278318701429624, −2.54405409308194698593413306208, −1.48417783057371004061027479295, −0.925558155661998755991364751695,
0.925558155661998755991364751695, 1.48417783057371004061027479295, 2.54405409308194698593413306208, 3.88961624172070278318701429624, 4.41283997663094533202593594826, 5.41180118180413728835689362746, 6.08710113366665458314298997586, 6.80320970906022861265370652393, 7.42831316722326391535076090152, 8.344879216440505112341823815918