| L(s) = 1 | − 4.55e6·2-s − 1.20e14·4-s + 3.08e16·5-s + 1.11e20·7-s + 1.18e21·8-s − 1.40e23·10-s + 1.19e24·11-s − 2.71e24·13-s − 5.05e26·14-s + 1.14e28·16-s − 1.17e29·17-s − 1.65e30·19-s − 3.70e30·20-s − 5.43e30·22-s − 5.41e31·23-s + 2.40e32·25-s + 1.23e31·26-s − 1.33e34·28-s + 1.03e34·29-s − 3.80e34·31-s − 2.19e35·32-s + 5.34e35·34-s + 3.42e36·35-s + 8.41e36·37-s + 7.53e36·38-s + 3.65e37·40-s − 7.68e37·41-s + ⋯ |
| L(s) = 1 | − 0.383·2-s − 0.852·4-s + 1.15·5-s + 1.53·7-s + 0.710·8-s − 0.443·10-s + 0.402·11-s − 0.0180·13-s − 0.588·14-s + 0.580·16-s − 1.42·17-s − 1.47·19-s − 0.986·20-s − 0.154·22-s − 0.541·23-s + 0.338·25-s + 0.00691·26-s − 1.30·28-s + 0.445·29-s − 0.341·31-s − 0.933·32-s + 0.547·34-s + 1.77·35-s + 1.18·37-s + 0.565·38-s + 0.822·40-s − 0.966·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{49}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 4.55e6T + 1.40e14T^{2} \) |
| 5 | \( 1 - 3.08e16T + 7.10e32T^{2} \) |
| 7 | \( 1 - 1.11e20T + 5.24e39T^{2} \) |
| 11 | \( 1 - 1.19e24T + 8.81e48T^{2} \) |
| 13 | \( 1 + 2.71e24T + 2.26e52T^{2} \) |
| 17 | \( 1 + 1.17e29T + 6.77e57T^{2} \) |
| 19 | \( 1 + 1.65e30T + 1.26e60T^{2} \) |
| 23 | \( 1 + 5.41e31T + 1.00e64T^{2} \) |
| 29 | \( 1 - 1.03e34T + 5.40e68T^{2} \) |
| 31 | \( 1 + 3.80e34T + 1.24e70T^{2} \) |
| 37 | \( 1 - 8.41e36T + 5.07e73T^{2} \) |
| 41 | \( 1 + 7.68e37T + 6.32e75T^{2} \) |
| 43 | \( 1 + 2.17e38T + 5.92e76T^{2} \) |
| 47 | \( 1 + 2.73e39T + 3.87e78T^{2} \) |
| 53 | \( 1 + 8.29e39T + 1.09e81T^{2} \) |
| 59 | \( 1 + 7.09e40T + 1.69e83T^{2} \) |
| 61 | \( 1 + 6.43e41T + 8.13e83T^{2} \) |
| 67 | \( 1 - 1.19e43T + 6.69e85T^{2} \) |
| 71 | \( 1 + 2.64e43T + 1.02e87T^{2} \) |
| 73 | \( 1 - 3.87e43T + 3.76e87T^{2} \) |
| 79 | \( 1 + 1.22e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 1.65e45T + 1.57e90T^{2} \) |
| 89 | \( 1 - 6.00e45T + 4.18e91T^{2} \) |
| 97 | \( 1 + 1.75e46T + 2.38e93T^{2} \) |
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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
−11.25994602917337394994276689931, −10.12569155295855028732123169635, −8.930064590899952729243834638731, −8.125916188414435032013473602073, −6.39444036117361291714390984275, −5.03119849712762214046151690135, −4.23607960387633193789318476807, −2.11034618492272821162541810220, −1.45214145236988996693042056990, 0,
1.45214145236988996693042056990, 2.11034618492272821162541810220, 4.23607960387633193789318476807, 5.03119849712762214046151690135, 6.39444036117361291714390984275, 8.125916188414435032013473602073, 8.930064590899952729243834638731, 10.12569155295855028732123169635, 11.25994602917337394994276689931
