| L(s) = 1 | + 4.19e6·2-s + 8.42e10·3-s + 1.75e13·4-s + 9.15e15·5-s + 3.53e17·6-s + 2.40e18·7-s + 7.37e19·8-s + 4.14e21·9-s + 3.83e22·10-s − 2.26e22·11-s + 1.48e24·12-s − 1.41e25·13-s + 1.00e25·14-s + 7.71e26·15-s + 3.09e26·16-s − 8.28e27·17-s + 1.73e28·18-s − 8.92e28·19-s + 1.60e29·20-s + 2.02e29·21-s − 9.51e28·22-s + 5.64e28·23-s + 6.21e30·24-s + 5.53e31·25-s − 5.94e31·26-s + 1.00e32·27-s + 4.23e31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.55·3-s + 0.5·4-s + 1.71·5-s + 1.09·6-s + 0.232·7-s + 0.353·8-s + 1.40·9-s + 1.21·10-s − 0.0839·11-s + 0.775·12-s − 1.22·13-s + 0.164·14-s + 2.66·15-s + 0.250·16-s − 1.71·17-s + 0.992·18-s − 1.50·19-s + 0.858·20-s + 0.360·21-s − 0.0593·22-s + 0.0129·23-s + 0.548·24-s + 1.94·25-s − 0.865·26-s + 0.624·27-s + 0.116·28-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
| \(L(23)\) |
\(\approx\) |
\(6.237109586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.237109586\) |
| \(L(\frac{47}{2})\) |
|
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 | \( 1 - 8.42e10T + 2.95e21T^{2} \) |
| 5 | \( 1 - 9.15e15T + 2.84e31T^{2} \) |
| 7 | \( 1 - 2.40e18T + 1.07e38T^{2} \) |
| 11 | \( 1 + 2.26e22T + 7.28e46T^{2} \) |
| 13 | \( 1 + 1.41e25T + 1.34e50T^{2} \) |
| 17 | \( 1 + 8.28e27T + 2.34e55T^{2} \) |
| 19 | \( 1 + 8.92e28T + 3.49e57T^{2} \) |
| 23 | \( 1 - 5.64e28T + 1.89e61T^{2} \) |
| 29 | \( 1 - 9.80e32T + 6.42e65T^{2} \) |
| 31 | \( 1 - 3.29e33T + 1.29e67T^{2} \) |
| 37 | \( 1 + 7.74e34T + 3.70e70T^{2} \) |
| 41 | \( 1 - 5.05e35T + 3.76e72T^{2} \) |
| 43 | \( 1 + 5.76e36T + 3.20e73T^{2} \) |
| 47 | \( 1 - 2.26e37T + 1.75e75T^{2} \) |
| 53 | \( 1 - 4.94e38T + 3.91e77T^{2} \) |
| 59 | \( 1 - 6.55e39T + 4.87e79T^{2} \) |
| 61 | \( 1 + 9.04e38T + 2.18e80T^{2} \) |
| 67 | \( 1 - 5.14e40T + 1.49e82T^{2} \) |
| 71 | \( 1 + 3.63e41T + 2.02e83T^{2} \) |
| 73 | \( 1 - 1.16e41T + 7.07e83T^{2} \) |
| 79 | \( 1 - 6.51e41T + 2.47e85T^{2} \) |
| 83 | \( 1 - 7.53e42T + 2.28e86T^{2} \) |
| 89 | \( 1 - 8.95e42T + 5.27e87T^{2} \) |
| 97 | \( 1 + 3.53e44T + 2.53e89T^{2} \) |
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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
−17.47181041753520800793968944156, −15.02998543214135205344843893413, −13.96916314935017875788821637951, −13.02465815304419359437668139808, −10.13036915167214385632441297663, −8.699816738451365361174609907510, −6.61719473087070625099122866796, −4.64959870563031222004499275381, −2.56767703649441236695384257121, −2.01162741302255434184756689084,
2.01162741302255434184756689084, 2.56767703649441236695384257121, 4.64959870563031222004499275381, 6.61719473087070625099122866796, 8.699816738451365361174609907510, 10.13036915167214385632441297663, 13.02465815304419359437668139808, 13.96916314935017875788821637951, 15.02998543214135205344843893413, 17.47181041753520800793968944156
