L(s) = 1 | − 1.08e4·2-s − 4.78e6·3-s − 4.19e8·4-s + 2.20e10·5-s + 5.19e10·6-s − 6.70e11·7-s + 1.03e13·8-s + 2.28e13·9-s − 2.38e14·10-s + 1.26e15·11-s + 2.00e15·12-s − 2.39e16·13-s + 7.28e15·14-s − 1.05e17·15-s + 1.12e17·16-s − 6.62e17·17-s − 2.48e17·18-s − 4.89e18·19-s − 9.21e18·20-s + 3.20e18·21-s − 1.37e19·22-s − 1.50e19·23-s − 4.96e19·24-s + 2.97e20·25-s + 2.60e20·26-s − 1.09e20·27-s + 2.81e20·28-s + ⋯ |
L(s) = 1 | − 0.468·2-s − 0.577·3-s − 0.780·4-s + 1.61·5-s + 0.270·6-s − 0.373·7-s + 0.834·8-s + 0.333·9-s − 0.755·10-s + 1.00·11-s + 0.450·12-s − 1.68·13-s + 0.175·14-s − 0.930·15-s + 0.389·16-s − 0.954·17-s − 0.156·18-s − 1.40·19-s − 1.25·20-s + 0.215·21-s − 0.471·22-s − 0.270·23-s − 0.481·24-s + 1.59·25-s + 0.790·26-s − 0.192·27-s + 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(15)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{31}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 4.78e6T \) |
good | 2 | \( 1 + 1.08e4T + 5.36e8T^{2} \) |
| 5 | \( 1 - 2.20e10T + 1.86e20T^{2} \) |
| 7 | \( 1 + 6.70e11T + 3.21e24T^{2} \) |
| 11 | \( 1 - 1.26e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 2.39e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 6.62e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 4.89e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 1.50e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 3.99e20T + 2.56e42T^{2} \) |
| 31 | \( 1 - 1.20e21T + 1.77e43T^{2} \) |
| 37 | \( 1 + 9.70e22T + 3.00e45T^{2} \) |
| 41 | \( 1 - 6.14e22T + 5.89e46T^{2} \) |
| 43 | \( 1 - 1.07e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + 1.01e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.63e25T + 1.00e50T^{2} \) |
| 59 | \( 1 + 1.58e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 3.52e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 1.44e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 4.44e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + 3.43e26T + 1.08e54T^{2} \) |
| 79 | \( 1 + 5.48e27T + 1.07e55T^{2} \) |
| 83 | \( 1 - 5.16e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 3.23e28T + 3.40e56T^{2} \) |
| 97 | \( 1 - 5.22e27T + 4.13e57T^{2} \) |
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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
−17.54325833699976025160366699611, −17.10835813026695821590247734818, −14.24786298158883391104484561517, −12.76971412813786345004273270846, −10.21597040378714655350991290085, −9.179276186295923644281376754626, −6.49060567627457641714563629186, −4.78248232069624325728104945710, −1.84917962322803316114029015551, 0,
1.84917962322803316114029015551, 4.78248232069624325728104945710, 6.49060567627457641714563629186, 9.179276186295923644281376754626, 10.21597040378714655350991290085, 12.76971412813786345004273270846, 14.24786298158883391104484561517, 17.10835813026695821590247734818, 17.54325833699976025160366699611