| L(s) = 1 | + 19.4·2-s + 36.3·3-s + 249.·4-s − 31.9·5-s + 706.·6-s − 461.·7-s + 2.35e3·8-s − 864.·9-s − 620.·10-s + 4.54e3·11-s + 9.06e3·12-s − 1.11e4·13-s − 8.95e3·14-s − 1.16e3·15-s + 1.38e4·16-s + 1.45e4·17-s − 1.67e4·18-s + 2.07e4·19-s − 7.96e3·20-s − 1.67e4·21-s + 8.81e4·22-s − 1.21e4·23-s + 8.56e4·24-s − 7.71e4·25-s − 2.17e5·26-s − 1.10e5·27-s − 1.14e5·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.777·3-s + 1.94·4-s − 0.114·5-s + 1.33·6-s − 0.508·7-s + 1.62·8-s − 0.395·9-s − 0.196·10-s + 1.02·11-s + 1.51·12-s − 1.41·13-s − 0.872·14-s − 0.0888·15-s + 0.845·16-s + 0.717·17-s − 0.678·18-s + 0.693·19-s − 0.222·20-s − 0.395·21-s + 1.76·22-s − 0.208·23-s + 1.26·24-s − 0.986·25-s − 2.42·26-s − 1.08·27-s − 0.989·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.583349598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.583349598\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 19.4T + 128T^{2} \) |
| 3 | \( 1 - 36.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 31.9T + 7.81e4T^{2} \) |
| 7 | \( 1 + 461.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.11e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.07e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 3.16e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.55e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.79e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.00e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.52e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.39e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.79e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.62e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.45e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.71e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.85e6T + 8.07e13T^{2} \) |
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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
−15.69834523726965193388533133295, −14.49433609207500576560398071993, −14.02763148363492491918433297327, −12.57182672942102068240469826137, −11.63108378947355486101266159681, −9.508442595373346478398970467678, −7.39884499895639537174001564447, −5.74578234666367604545040406338, −3.94216953044909167995387877567, −2.62159998945481565999424197818,
2.62159998945481565999424197818, 3.94216953044909167995387877567, 5.74578234666367604545040406338, 7.39884499895639537174001564447, 9.508442595373346478398970467678, 11.63108378947355486101266159681, 12.57182672942102068240469826137, 14.02763148363492491918433297327, 14.49433609207500576560398071993, 15.69834523726965193388533133295
