| L(s) = 1 | − 2.12e3·2-s − 5.90e4·3-s + 2.42e6·4-s + 1.25e8·6-s + 1.04e9·7-s − 7.05e8·8-s + 3.48e9·9-s + 2.38e10·11-s − 1.43e11·12-s − 3.64e11·13-s − 2.21e12·14-s − 3.59e12·16-s + 8.45e12·17-s − 7.41e12·18-s + 5.34e11·19-s − 6.15e13·21-s − 5.07e13·22-s − 3.10e14·23-s + 4.16e13·24-s + 7.76e14·26-s − 2.05e14·27-s + 2.53e15·28-s + 1.78e15·29-s + 6.32e15·31-s + 9.12e15·32-s − 1.40e15·33-s − 1.79e16·34-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 0.577·3-s + 1.15·4-s + 0.848·6-s + 1.39·7-s − 0.232·8-s + 0.333·9-s + 0.277·11-s − 0.668·12-s − 0.734·13-s − 2.04·14-s − 0.816·16-s + 1.01·17-s − 0.489·18-s + 0.0200·19-s − 0.804·21-s − 0.407·22-s − 1.56·23-s + 0.134·24-s + 1.07·26-s − 0.192·27-s + 1.61·28-s + 0.789·29-s + 1.38·31-s + 1.43·32-s − 0.160·33-s − 1.49·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.027541148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027541148\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.12e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 1.04e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 2.38e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.64e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.45e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 5.34e11T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.10e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.78e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.32e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.11e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 2.38e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.06e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 6.65e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.12e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 3.51e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.16e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.34e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.06e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.09e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 3.58e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.25e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 4.16e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 4.65e20T + 5.27e41T^{2} \) |
| 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
−10.35715840435849586587929092012, −9.753093928811493009436927592301, −8.318488210568514945167541772747, −7.83671682622139503587752297648, −6.67077731210145663806152258565, −5.28230571223925078693961318596, −4.26585583988635728955490276017, −2.32832395440030470039805838238, −1.37275454631632053781901796651, −0.60786254441458650013198837257,
0.60786254441458650013198837257, 1.37275454631632053781901796651, 2.32832395440030470039805838238, 4.26585583988635728955490276017, 5.28230571223925078693961318596, 6.67077731210145663806152258565, 7.83671682622139503587752297648, 8.318488210568514945167541772747, 9.753093928811493009436927592301, 10.35715840435849586587929092012
