| L(s) = 1 | + 938.·2-s − 1.56e5·3-s − 1.21e6·4-s − 1.46e8·6-s + 2.53e8·7-s − 3.11e9·8-s + 1.39e10·9-s − 8.13e10·11-s + 1.89e11·12-s + 3.79e11·13-s + 2.38e11·14-s − 3.71e11·16-s + 9.55e12·17-s + 1.30e13·18-s − 1.33e13·19-s − 3.96e13·21-s − 7.63e13·22-s + 3.84e14·23-s + 4.85e14·24-s + 3.56e14·26-s − 5.37e14·27-s − 3.08e14·28-s + 4.23e15·29-s − 6.80e15·31-s + 6.17e15·32-s + 1.26e16·33-s + 8.97e15·34-s + ⋯ |
| L(s) = 1 | + 0.648·2-s − 1.52·3-s − 0.579·4-s − 0.989·6-s + 0.339·7-s − 1.02·8-s + 1.32·9-s − 0.945·11-s + 0.884·12-s + 0.763·13-s + 0.220·14-s − 0.0844·16-s + 1.14·17-s + 0.861·18-s − 0.500·19-s − 0.518·21-s − 0.612·22-s + 1.93·23-s + 1.56·24-s + 0.494·26-s − 0.502·27-s − 0.196·28-s + 1.86·29-s − 1.49·31-s + 0.969·32-s + 1.44·33-s + 0.745·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 938.T + 2.09e6T^{2} \) |
| 3 | \( 1 + 1.56e5T + 1.04e10T^{2} \) |
| 7 | \( 1 - 2.53e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 8.13e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 3.79e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 9.55e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.33e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.84e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 4.23e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 6.80e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.29e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.24e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.83e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.37e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.51e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 4.47e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 2.42e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 5.38e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.93e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.84e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.76e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 7.76e17T + 1.99e40T^{2} \) |
| 89 | \( 1 - 6.27e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 5.50e20T + 5.27e41T^{2} \) |
| 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
−12.40667761358222537010741309354, −11.25208462508164626944160274515, −10.19663092206303719149085612120, −8.461566897430610487452134461170, −6.66244649029312466930006252983, −5.37872670167196422277967481439, −4.90923310052312400716749777976, −3.31678297679475190129143432461, −1.11346979200934604622646026329, 0,
1.11346979200934604622646026329, 3.31678297679475190129143432461, 4.90923310052312400716749777976, 5.37872670167196422277967481439, 6.66244649029312466930006252983, 8.461566897430610487452134461170, 10.19663092206303719149085612120, 11.25208462508164626944160274515, 12.40667761358222537010741309354
