| L(s) = 1 | + 3.62e6·2-s + 1.01e10·3-s + 4.35e12·4-s + 6.19e14·5-s + 3.68e16·6-s + 2.58e18·7-s − 1.60e19·8-s − 2.25e20·9-s + 2.24e21·10-s + 2.74e22·11-s + 4.42e22·12-s + 6.62e23·13-s + 9.38e24·14-s + 6.29e24·15-s − 9.67e25·16-s + 5.82e25·17-s − 8.16e26·18-s − 1.98e27·19-s + 2.70e27·20-s + 2.62e28·21-s + 9.94e28·22-s − 2.61e29·23-s − 1.63e29·24-s − 7.52e29·25-s + 2.40e30·26-s − 5.61e30·27-s + 1.12e31·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.560·3-s + 0.495·4-s + 0.581·5-s + 0.685·6-s + 1.75·7-s − 0.616·8-s − 0.686·9-s + 0.710·10-s + 1.11·11-s + 0.277·12-s + 0.743·13-s + 2.14·14-s + 0.325·15-s − 1.24·16-s + 0.204·17-s − 0.839·18-s − 0.636·19-s + 0.288·20-s + 0.980·21-s + 1.36·22-s − 1.37·23-s − 0.345·24-s − 0.662·25-s + 0.909·26-s − 0.944·27-s + 0.867·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(44-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+43/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(4.325994499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.325994499\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 3.62e6T + 8.79e12T^{2} \) |
| 3 | \( 1 - 1.01e10T + 3.28e20T^{2} \) |
| 5 | \( 1 - 6.19e14T + 1.13e30T^{2} \) |
| 7 | \( 1 - 2.58e18T + 2.18e36T^{2} \) |
| 11 | \( 1 - 2.74e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 6.62e23T + 7.93e47T^{2} \) |
| 17 | \( 1 - 5.82e25T + 8.11e52T^{2} \) |
| 19 | \( 1 + 1.98e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 2.61e29T + 3.58e58T^{2} \) |
| 29 | \( 1 - 2.27e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 1.73e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 5.95e32T + 2.70e67T^{2} \) |
| 41 | \( 1 + 5.56e34T + 2.23e69T^{2} \) |
| 43 | \( 1 - 1.13e35T + 1.73e70T^{2} \) |
| 47 | \( 1 - 2.48e35T + 7.94e71T^{2} \) |
| 53 | \( 1 - 3.93e36T + 1.39e74T^{2} \) |
| 59 | \( 1 - 6.06e37T + 1.40e76T^{2} \) |
| 61 | \( 1 - 3.31e38T + 5.87e76T^{2} \) |
| 67 | \( 1 - 2.46e38T + 3.32e78T^{2} \) |
| 71 | \( 1 + 5.22e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 1.32e40T + 1.32e80T^{2} \) |
| 79 | \( 1 + 1.99e40T + 3.96e81T^{2} \) |
| 83 | \( 1 + 5.56e39T + 3.31e82T^{2} \) |
| 89 | \( 1 + 2.64e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 3.46e42T + 2.69e85T^{2} \) |
| 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
−21.78623200245439295075697197117, −20.51071314133364406788961231168, −17.70121612920923324856198162370, −14.64298428300446705778482044168, −13.90770841434162502850369730943, −11.62630428838795880795745896470, −8.611581403957278931214567809049, −5.74213419822693390400659725987, −4.00583767473095125197890886134, −1.93411219675686423614527871698,
1.93411219675686423614527871698, 4.00583767473095125197890886134, 5.74213419822693390400659725987, 8.611581403957278931214567809049, 11.62630428838795880795745896470, 13.90770841434162502850369730943, 14.64298428300446705778482044168, 17.70121612920923324856198162370, 20.51071314133364406788961231168, 21.78623200245439295075697197117
