| L(s) = 1 | − 4.19e6·2-s − 1.07e11·3-s + 1.75e13·4-s − 5.35e15·5-s + 4.52e17·6-s − 5.31e18·7-s − 7.37e19·8-s + 8.67e21·9-s + 2.24e22·10-s + 3.01e23·11-s − 1.89e24·12-s − 3.58e24·13-s + 2.22e25·14-s + 5.77e26·15-s + 3.09e26·16-s + 2.22e26·17-s − 3.64e28·18-s + 4.98e28·19-s − 9.41e28·20-s + 5.73e29·21-s − 1.26e30·22-s − 4.72e30·23-s + 7.95e30·24-s + 2.11e29·25-s + 1.50e31·26-s − 6.17e32·27-s − 9.35e31·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.98·3-s + 0.5·4-s − 1.00·5-s + 1.40·6-s − 0.513·7-s − 0.353·8-s + 2.93·9-s + 0.709·10-s + 1.11·11-s − 0.992·12-s − 0.309·13-s + 0.363·14-s + 1.99·15-s + 0.250·16-s + 0.0459·17-s − 2.07·18-s + 0.842·19-s − 0.501·20-s + 1.01·21-s − 0.788·22-s − 1.08·23-s + 0.701·24-s + 0.00743·25-s + 0.218·26-s − 3.84·27-s − 0.256·28-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
| \(L(23)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{47}{2})\) |
|
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 | \( 1 + 1.07e11T + 2.95e21T^{2} \) |
| 5 | \( 1 + 5.35e15T + 2.84e31T^{2} \) |
| 7 | \( 1 + 5.31e18T + 1.07e38T^{2} \) |
| 11 | \( 1 - 3.01e23T + 7.28e46T^{2} \) |
| 13 | \( 1 + 3.58e24T + 1.34e50T^{2} \) |
| 17 | \( 1 - 2.22e26T + 2.34e55T^{2} \) |
| 19 | \( 1 - 4.98e28T + 3.49e57T^{2} \) |
| 23 | \( 1 + 4.72e30T + 1.89e61T^{2} \) |
| 29 | \( 1 - 6.15e32T + 6.42e65T^{2} \) |
| 31 | \( 1 - 5.25e32T + 1.29e67T^{2} \) |
| 37 | \( 1 - 2.05e35T + 3.70e70T^{2} \) |
| 41 | \( 1 - 1.88e36T + 3.76e72T^{2} \) |
| 43 | \( 1 + 1.70e36T + 3.20e73T^{2} \) |
| 47 | \( 1 - 3.61e37T + 1.75e75T^{2} \) |
| 53 | \( 1 - 9.38e38T + 3.91e77T^{2} \) |
| 59 | \( 1 + 4.98e39T + 4.87e79T^{2} \) |
| 61 | \( 1 - 2.73e39T + 2.18e80T^{2} \) |
| 67 | \( 1 + 1.07e41T + 1.49e82T^{2} \) |
| 71 | \( 1 + 9.82e40T + 2.02e83T^{2} \) |
| 73 | \( 1 - 2.35e41T + 7.07e83T^{2} \) |
| 79 | \( 1 + 9.11e42T + 2.47e85T^{2} \) |
| 83 | \( 1 + 3.78e42T + 2.28e86T^{2} \) |
| 89 | \( 1 + 8.23e43T + 5.27e87T^{2} \) |
| 97 | \( 1 + 1.69e44T + 2.53e89T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.70792553303538617188813378640, −15.76705889999840112410335929169, −12.20938758271397067892753809098, −11.44847503671874639495569721298, −9.893910018844672814393393700663, −7.31887357579733563461257722515, −6.05937964206145310528332945306, −4.16888671496806883989512136484, −1.06307248070571921365479082993, 0,
1.06307248070571921365479082993, 4.16888671496806883989512136484, 6.05937964206145310528332945306, 7.31887357579733563461257722515, 9.893910018844672814393393700663, 11.44847503671874639495569721298, 12.20938758271397067892753809098, 15.76705889999840112410335929169, 16.70792553303538617188813378640
