L(s) = 1 | + 1.29e15·2-s − 2.26e24·3-s − 8.55e29·4-s − 1.52e35·5-s − 2.93e39·6-s − 2.30e42·7-s − 4.39e45·8-s + 3.57e48·9-s − 1.98e50·10-s + 7.37e52·11-s + 1.93e54·12-s + 1.08e56·13-s − 2.98e57·14-s + 3.46e59·15-s − 3.52e60·16-s − 7.80e61·17-s + 4.63e63·18-s − 2.75e64·19-s + 1.30e65·20-s + 5.21e66·21-s + 9.56e67·22-s + 8.02e68·23-s + 9.94e69·24-s − 1.60e70·25-s + 1.40e71·26-s − 4.59e72·27-s + 1.97e72·28-s + ⋯ |
L(s) = 1 | + 0.813·2-s − 1.82·3-s − 0.337·4-s − 0.770·5-s − 1.48·6-s − 0.483·7-s − 1.08·8-s + 2.31·9-s − 0.626·10-s + 1.89·11-s + 0.614·12-s + 0.602·13-s − 0.393·14-s + 1.40·15-s − 0.548·16-s − 0.568·17-s + 1.88·18-s − 0.729·19-s + 0.260·20-s + 0.880·21-s + 1.54·22-s + 1.37·23-s + 1.98·24-s − 0.406·25-s + 0.490·26-s − 2.38·27-s + 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(51)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{103}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 - 1.29e15T + 2.53e30T^{2} \) |
| 3 | \( 1 + 2.26e24T + 1.54e48T^{2} \) |
| 5 | \( 1 + 1.52e35T + 3.94e70T^{2} \) |
| 7 | \( 1 + 2.30e42T + 2.26e85T^{2} \) |
| 11 | \( 1 - 7.37e52T + 1.51e105T^{2} \) |
| 13 | \( 1 - 1.08e56T + 3.22e112T^{2} \) |
| 17 | \( 1 + 7.80e61T + 1.88e124T^{2} \) |
| 19 | \( 1 + 2.75e64T + 1.42e129T^{2} \) |
| 23 | \( 1 - 8.02e68T + 3.42e137T^{2} \) |
| 29 | \( 1 + 4.17e73T + 5.03e147T^{2} \) |
| 31 | \( 1 - 1.32e74T + 4.24e150T^{2} \) |
| 37 | \( 1 - 7.61e78T + 2.44e158T^{2} \) |
| 41 | \( 1 - 4.16e81T + 7.78e162T^{2} \) |
| 43 | \( 1 + 6.23e81T + 9.55e164T^{2} \) |
| 47 | \( 1 + 9.39e83T + 7.61e168T^{2} \) |
| 53 | \( 1 - 1.74e87T + 1.41e174T^{2} \) |
| 59 | \( 1 - 2.25e89T + 7.17e178T^{2} \) |
| 61 | \( 1 + 2.74e90T + 2.08e180T^{2} \) |
| 67 | \( 1 - 8.52e91T + 2.71e184T^{2} \) |
| 71 | \( 1 + 3.72e93T + 9.48e186T^{2} \) |
| 73 | \( 1 + 2.87e93T + 1.56e188T^{2} \) |
| 79 | \( 1 - 4.49e95T + 4.57e191T^{2} \) |
| 83 | \( 1 - 1.36e96T + 6.71e193T^{2} \) |
| 89 | \( 1 - 2.37e98T + 7.73e196T^{2} \) |
| 97 | \( 1 + 7.96e99T + 4.61e200T^{2} \) |
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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
−13.07671410039418899376880312522, −11.99544613062561610326758848976, −11.14342480777275099157871844788, −9.214437841642332248701727031610, −6.75363602026425118950790788001, −5.94131937938569002000718710016, −4.49509318553229487084691255763, −3.78987939913398560764337013948, −1.04157930115495124271021096389, 0,
1.04157930115495124271021096389, 3.78987939913398560764337013948, 4.49509318553229487084691255763, 5.94131937938569002000718710016, 6.75363602026425118950790788001, 9.214437841642332248701727031610, 11.14342480777275099157871844788, 11.99544613062561610326758848976, 13.07671410039418899376880312522