| L(s) = 1 | + 1.63e4·2-s − 4.78e6·3-s + 2.68e8·4-s − 2.03e10·5-s − 7.83e10·6-s − 2.52e12·7-s + 4.39e12·8-s + 2.28e13·9-s − 3.34e14·10-s + 2.16e15·11-s − 1.28e15·12-s − 1.73e16·13-s − 4.13e16·14-s + 9.75e16·15-s + 7.20e16·16-s + 1.09e18·17-s + 3.74e17·18-s + 1.44e18·19-s − 5.47e18·20-s + 1.20e19·21-s + 3.55e19·22-s − 7.43e19·23-s − 2.10e19·24-s + 2.29e20·25-s − 2.83e20·26-s − 1.09e20·27-s − 6.76e20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.49·5-s − 0.408·6-s − 1.40·7-s + 0.353·8-s + 0.333·9-s − 1.05·10-s + 1.72·11-s − 0.288·12-s − 1.21·13-s − 0.993·14-s + 0.862·15-s + 0.250·16-s + 1.57·17-s + 0.235·18-s + 0.413·19-s − 0.746·20-s + 0.811·21-s + 1.21·22-s − 1.33·23-s − 0.204·24-s + 1.23·25-s − 0.862·26-s − 0.192·27-s − 0.702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.525215756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525215756\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.63e4T \) |
| 3 | \( 1 + 4.78e6T \) |
| good | 5 | \( 1 + 2.03e10T + 1.86e20T^{2} \) |
| 7 | \( 1 + 2.52e12T + 3.21e24T^{2} \) |
| 11 | \( 1 - 2.16e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 1.73e16T + 2.01e32T^{2} \) |
| 17 | \( 1 - 1.09e18T + 4.81e35T^{2} \) |
| 19 | \( 1 - 1.44e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 7.43e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 1.78e21T + 2.56e42T^{2} \) |
| 31 | \( 1 - 1.83e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 6.04e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 7.54e22T + 5.89e46T^{2} \) |
| 43 | \( 1 - 4.59e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 7.22e23T + 3.09e48T^{2} \) |
| 53 | \( 1 + 5.39e24T + 1.00e50T^{2} \) |
| 59 | \( 1 + 6.59e24T + 2.26e51T^{2} \) |
| 61 | \( 1 - 6.51e24T + 5.95e51T^{2} \) |
| 67 | \( 1 + 1.98e26T + 9.04e52T^{2} \) |
| 71 | \( 1 + 4.25e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + 4.61e26T + 1.08e54T^{2} \) |
| 79 | \( 1 - 5.25e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 4.10e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + 5.01e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 9.02e28T + 4.13e57T^{2} \) |
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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
−15.89250785544616488085240959992, −14.43223730886219405482338595772, −12.25443177406686536315513418024, −11.91760671382680273597192390170, −9.844530996312542117069765897938, −7.47461678483791941303538645873, −6.22247897423420474833625498813, −4.29217833923208007065125030145, −3.26443376813767325696367585896, −0.70679210904093464877437853522,
0.70679210904093464877437853522, 3.26443376813767325696367585896, 4.29217833923208007065125030145, 6.22247897423420474833625498813, 7.47461678483791941303538645873, 9.844530996312542117069765897938, 11.91760671382680273597192390170, 12.25443177406686536315513418024, 14.43223730886219405482338595772, 15.89250785544616488085240959992
