L(s) = 1 | + 2-s + 2.32·3-s + 4-s + 1.93·5-s + 2.32·6-s + 2.88·7-s + 8-s + 2.40·9-s + 1.93·10-s + 1.33·11-s + 2.32·12-s + 2.89·13-s + 2.88·14-s + 4.50·15-s + 16-s + 5.30·17-s + 2.40·18-s + 19-s + 1.93·20-s + 6.70·21-s + 1.33·22-s − 5.19·23-s + 2.32·24-s − 1.24·25-s + 2.89·26-s − 1.37·27-s + 2.88·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.34·3-s + 0.5·4-s + 0.866·5-s + 0.949·6-s + 1.08·7-s + 0.353·8-s + 0.803·9-s + 0.612·10-s + 0.402·11-s + 0.671·12-s + 0.803·13-s + 0.770·14-s + 1.16·15-s + 0.250·16-s + 1.28·17-s + 0.567·18-s + 0.229·19-s + 0.433·20-s + 1.46·21-s + 0.284·22-s − 1.08·23-s + 0.474·24-s − 0.249·25-s + 0.568·26-s − 0.264·27-s + 0.544·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.087533108\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.087533108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 5 | \( 1 - 1.93T + 5T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 3.02T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 + 5.56T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 0.276T + 43T^{2} \) |
| 47 | \( 1 - 0.439T + 47T^{2} \) |
| 53 | \( 1 - 7.22T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.972T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.82496742615151635450523039611, −7.37117956687622992281837424954, −6.19796618159526998939112681439, −5.81191437930284966864913798293, −4.97532776804384379588627288486, −4.13942642907152369770854197180, −3.47205897637348550589144741155, −2.77579671478030625618154193281, −1.77162124680392699471205056025, −1.47902558839117623134852902831,
1.47902558839117623134852902831, 1.77162124680392699471205056025, 2.77579671478030625618154193281, 3.47205897637348550589144741155, 4.13942642907152369770854197180, 4.97532776804384379588627288486, 5.81191437930284966864913798293, 6.19796618159526998939112681439, 7.37117956687622992281837424954, 7.82496742615151635450523039611