| L(s) = 1 | − 2-s + 0.493·3-s + 4-s + 0.378·5-s − 0.493·6-s + 2.20·7-s − 8-s − 2.75·9-s − 0.378·10-s + 4.52·11-s + 0.493·12-s + 1.19·13-s − 2.20·14-s + 0.187·15-s + 16-s − 3.33·17-s + 2.75·18-s + 4.40·19-s + 0.378·20-s + 1.08·21-s − 4.52·22-s + 3.28·23-s − 0.493·24-s − 4.85·25-s − 1.19·26-s − 2.84·27-s + 2.20·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.285·3-s + 0.5·4-s + 0.169·5-s − 0.201·6-s + 0.833·7-s − 0.353·8-s − 0.918·9-s − 0.119·10-s + 1.36·11-s + 0.142·12-s + 0.332·13-s − 0.589·14-s + 0.0483·15-s + 0.250·16-s − 0.809·17-s + 0.649·18-s + 1.01·19-s + 0.0847·20-s + 0.237·21-s − 0.965·22-s + 0.684·23-s − 0.100·24-s − 0.971·25-s − 0.234·26-s − 0.546·27-s + 0.416·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582600145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582600145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.493T + 3T^{2} \) |
| 5 | \( 1 - 0.378T + 5T^{2} \) |
| 7 | \( 1 - 2.20T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 - 3.28T + 23T^{2} \) |
| 29 | \( 1 - 0.343T + 29T^{2} \) |
| 31 | \( 1 - 8.98T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 0.516T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 59 | \( 1 + 1.09T + 59T^{2} \) |
| 61 | \( 1 - 6.20T + 61T^{2} \) |
| 67 | \( 1 - 8.54T + 67T^{2} \) |
| 71 | \( 1 + 4.35T + 71T^{2} \) |
| 73 | \( 1 + 5.26T + 73T^{2} \) |
| 79 | \( 1 + 0.376T + 79T^{2} \) |
| 83 | \( 1 + 4.33T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 + 1.08T + 97T^{2} \) |
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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
−9.326342950780281166999543838808, −8.657938243881355722064189853639, −8.163679863491607381065943085376, −7.16034861793970865363694740657, −6.35689714169034653950174021950, −5.49635644493570298589477389986, −4.36900402679907421487656874048, −3.26843809408895104567353688861, −2.15066866483411725892582738679, −1.04106477096327342081228800385,
1.04106477096327342081228800385, 2.15066866483411725892582738679, 3.26843809408895104567353688861, 4.36900402679907421487656874048, 5.49635644493570298589477389986, 6.35689714169034653950174021950, 7.16034861793970865363694740657, 8.163679863491607381065943085376, 8.657938243881355722064189853639, 9.326342950780281166999543838808
