| L(s) = 1 | + 4.19e6·2-s − 2.43e10·3-s + 1.75e13·4-s − 4.75e15·5-s − 1.02e17·6-s + 5.49e18·7-s + 7.37e19·8-s − 2.35e21·9-s − 1.99e22·10-s + 2.16e23·11-s − 4.29e23·12-s + 1.18e25·13-s + 2.30e25·14-s + 1.16e26·15-s + 3.09e26·16-s + 1.71e27·17-s − 9.89e27·18-s − 7.20e26·19-s − 8.36e28·20-s − 1.34e29·21-s + 9.06e29·22-s + 6.93e30·23-s − 1.80e30·24-s − 5.81e30·25-s + 4.97e31·26-s + 1.29e32·27-s + 9.67e31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.448·3-s + 0.5·4-s − 0.891·5-s − 0.317·6-s + 0.531·7-s + 0.353·8-s − 0.798·9-s − 0.630·10-s + 0.800·11-s − 0.224·12-s + 1.02·13-s + 0.375·14-s + 0.400·15-s + 0.250·16-s + 0.353·17-s − 0.564·18-s − 0.0121·19-s − 0.445·20-s − 0.238·21-s + 0.565·22-s + 1.59·23-s − 0.158·24-s − 0.204·25-s + 0.724·26-s + 0.807·27-s + 0.265·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(46-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+45/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(23)\) |
\(\approx\) |
\(2.460092122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.460092122\) |
| \(L(\frac{47}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.19e6T \) |
| good | 3 | \( 1 + 2.43e10T + 2.95e21T^{2} \) |
| 5 | \( 1 + 4.75e15T + 2.84e31T^{2} \) |
| 7 | \( 1 - 5.49e18T + 1.07e38T^{2} \) |
| 11 | \( 1 - 2.16e23T + 7.28e46T^{2} \) |
| 13 | \( 1 - 1.18e25T + 1.34e50T^{2} \) |
| 17 | \( 1 - 1.71e27T + 2.34e55T^{2} \) |
| 19 | \( 1 + 7.20e26T + 3.49e57T^{2} \) |
| 23 | \( 1 - 6.93e30T + 1.89e61T^{2} \) |
| 29 | \( 1 - 1.23e33T + 6.42e65T^{2} \) |
| 31 | \( 1 - 6.35e33T + 1.29e67T^{2} \) |
| 37 | \( 1 + 1.13e35T + 3.70e70T^{2} \) |
| 41 | \( 1 + 2.36e36T + 3.76e72T^{2} \) |
| 43 | \( 1 - 4.96e36T + 3.20e73T^{2} \) |
| 47 | \( 1 - 1.34e37T + 1.75e75T^{2} \) |
| 53 | \( 1 + 1.16e39T + 3.91e77T^{2} \) |
| 59 | \( 1 + 6.70e39T + 4.87e79T^{2} \) |
| 61 | \( 1 - 1.77e40T + 2.18e80T^{2} \) |
| 67 | \( 1 + 1.34e41T + 1.49e82T^{2} \) |
| 71 | \( 1 - 4.73e41T + 2.02e83T^{2} \) |
| 73 | \( 1 - 1.25e42T + 7.07e83T^{2} \) |
| 79 | \( 1 - 2.71e42T + 2.47e85T^{2} \) |
| 83 | \( 1 + 1.70e43T + 2.28e86T^{2} \) |
| 89 | \( 1 - 5.81e43T + 5.27e87T^{2} \) |
| 97 | \( 1 + 4.76e44T + 2.53e89T^{2} \) |
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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
−17.22155488884793515637000171613, −15.59728930414843338272652625574, −14.07006072227554698187630433643, −12.02335491260035703668570003039, −11.08295708198227584361145440251, −8.339124232919859230619858525948, −6.43773033268690564252243006983, −4.76977040760399375733466440754, −3.25545818777260741480786656395, −1.00239020039170817281045207556,
1.00239020039170817281045207556, 3.25545818777260741480786656395, 4.76977040760399375733466440754, 6.43773033268690564252243006983, 8.339124232919859230619858525948, 11.08295708198227584361145440251, 12.02335491260035703668570003039, 14.07006072227554698187630433643, 15.59728930414843338272652625574, 17.22155488884793515637000171613
