| L(s) = 1 | − 998.·2-s − 5.90e4·3-s − 1.09e6·4-s + 5.89e7·6-s − 1.20e9·7-s + 3.19e9·8-s + 3.48e9·9-s + 4.80e10·11-s + 6.49e10·12-s + 1.71e11·13-s + 1.20e12·14-s − 8.82e11·16-s + 1.14e13·17-s − 3.48e12·18-s + 7.38e12·19-s + 7.14e13·21-s − 4.79e13·22-s + 2.65e14·23-s − 1.88e14·24-s − 1.71e14·26-s − 2.05e14·27-s + 1.33e15·28-s + 3.36e14·29-s + 2.12e15·31-s − 5.81e15·32-s − 2.83e15·33-s − 1.14e16·34-s + ⋯ |
| L(s) = 1 | − 0.689·2-s − 0.577·3-s − 0.524·4-s + 0.398·6-s − 1.61·7-s + 1.05·8-s + 0.333·9-s + 0.558·11-s + 0.302·12-s + 0.344·13-s + 1.11·14-s − 0.200·16-s + 1.38·17-s − 0.229·18-s + 0.276·19-s + 0.934·21-s − 0.385·22-s + 1.33·23-s − 0.606·24-s − 0.237·26-s − 0.192·27-s + 0.848·28-s + 0.148·29-s + 0.465·31-s − 0.912·32-s − 0.322·33-s − 0.953·34-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
| \(L(11)\) |
\(\approx\) |
\(0.9634315210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9634315210\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 998.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.20e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 4.80e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 1.71e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.14e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 7.38e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.65e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.36e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.12e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 5.00e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 8.46e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 5.45e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.16e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 7.54e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 6.39e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 5.01e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.05e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.86e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.26e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.64e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 3.17e18T + 1.99e40T^{2} \) |
| 89 | \( 1 - 8.25e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.37e21T + 5.27e41T^{2} \) |
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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
−10.22875192262065347910040460440, −9.700052729088499641375981499059, −8.758436499692653881379191233177, −7.41261649199822110826887834568, −6.41949848338310847291750224001, −5.34797393581523395509139404496, −4.01939832295677867925029967116, −3.03179434059888051181839807531, −1.17802708886965043367788284758, −0.57451594454588857263231710700,
0.57451594454588857263231710700, 1.17802708886965043367788284758, 3.03179434059888051181839807531, 4.01939832295677867925029967116, 5.34797393581523395509139404496, 6.41949848338310847291750224001, 7.41261649199822110826887834568, 8.758436499692653881379191233177, 9.700052729088499641375981499059, 10.22875192262065347910040460440
