| L(s) = 1 | − 500.·2-s + 6.56e3·3-s + 1.19e5·4-s − 3.90e5·5-s − 3.28e6·6-s + 8.90e6·7-s + 5.87e6·8-s + 4.30e7·9-s + 1.95e8·10-s + 4.48e8·11-s + 7.82e8·12-s − 3.89e9·13-s − 4.45e9·14-s − 2.56e9·15-s − 1.85e10·16-s − 5.96e8·17-s − 2.15e10·18-s − 4.36e10·19-s − 4.66e10·20-s + 5.84e10·21-s − 2.24e11·22-s + 7.23e10·23-s + 3.85e10·24-s + 1.52e11·25-s + 1.95e12·26-s + 2.82e11·27-s + 1.06e12·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.577·3-s + 0.910·4-s − 0.447·5-s − 0.798·6-s + 0.583·7-s + 0.123·8-s + 0.333·9-s + 0.618·10-s + 0.631·11-s + 0.525·12-s − 1.32·13-s − 0.807·14-s − 0.258·15-s − 1.08·16-s − 0.0207·17-s − 0.460·18-s − 0.590·19-s − 0.407·20-s + 0.337·21-s − 0.872·22-s + 0.192·23-s + 0.0714·24-s + 0.200·25-s + 1.83·26-s + 0.192·27-s + 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.056966008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056966008\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 + 500.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 8.90e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 4.48e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.89e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 5.96e8T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.36e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 7.23e10T + 1.41e23T^{2} \) |
| 29 | \( 1 - 1.82e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 5.27e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.62e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 9.78e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.46e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.43e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.84e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 9.83e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.84e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.46e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.43e14T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.15e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.58e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.62e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 4.03e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.85e16T + 5.95e33T^{2} \) |
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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
−15.35783592593837813214671950907, −14.08364032134097829341674046913, −12.10985801368449589352488293925, −10.58956645878914093137388912657, −9.287642932894397584460434282228, −8.189591753603452565588249196003, −7.10564872257518644374861207282, −4.44596380633673329936632221632, −2.30248128046020392859577468076, −0.802194415274953505666270138771,
0.802194415274953505666270138771, 2.30248128046020392859577468076, 4.44596380633673329936632221632, 7.10564872257518644374861207282, 8.189591753603452565588249196003, 9.287642932894397584460434282228, 10.58956645878914093137388912657, 12.10985801368449589352488293925, 14.08364032134097829341674046913, 15.35783592593837813214671950907
