| L(s) = 1 | − 1.42·2-s + 0.0352·4-s + 0.964·5-s + 2.45·7-s + 2.80·8-s − 1.37·10-s − 2.64·11-s + 5.45·13-s − 3.50·14-s − 4.06·16-s − 4.34·17-s − 4.82·19-s + 0.0339·20-s + 3.77·22-s − 8.49·23-s − 4.06·25-s − 7.78·26-s + 0.0865·28-s + 1.53·29-s + 0.262·31-s + 0.199·32-s + 6.19·34-s + 2.37·35-s − 7.57·37-s + 6.88·38-s + 2.70·40-s − 4.20·41-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.0176·4-s + 0.431·5-s + 0.928·7-s + 0.991·8-s − 0.435·10-s − 0.797·11-s + 1.51·13-s − 0.936·14-s − 1.01·16-s − 1.05·17-s − 1.10·19-s + 0.00759·20-s + 0.804·22-s − 1.77·23-s − 0.813·25-s − 1.52·26-s + 0.0163·28-s + 0.284·29-s + 0.0471·31-s + 0.0352·32-s + 1.06·34-s + 0.400·35-s − 1.24·37-s + 1.11·38-s + 0.427·40-s − 0.656·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 + 1.42T + 2T^{2} \) |
| 5 | \( 1 - 0.964T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 + 2.64T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 8.49T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 - 0.262T + 31T^{2} \) |
| 37 | \( 1 + 7.57T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 0.544T + 53T^{2} \) |
| 59 | \( 1 + 3.68T + 59T^{2} \) |
| 61 | \( 1 + 5.22T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 - 5.17T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.452784323312441568718393981530, −8.341005395325243763866621141852, −7.44136555759383600869572085061, −6.38177317002744187092391205717, −5.62552540723555939599434343096, −4.56808942517811636864747302100, −3.90741202894576600469945614319, −2.20938165111073259882655413448, −1.55941209537754142741167736179, 0,
1.55941209537754142741167736179, 2.20938165111073259882655413448, 3.90741202894576600469945614319, 4.56808942517811636864747302100, 5.62552540723555939599434343096, 6.38177317002744187092391205717, 7.44136555759383600869572085061, 8.341005395325243763866621141852, 8.452784323312441568718393981530
