| L(s) = 1 | + 1.46e3·2-s − 5.90e4·3-s + 6.04e4·4-s − 8.67e7·6-s + 1.30e9·7-s − 2.99e9·8-s + 3.48e9·9-s + 1.42e11·11-s − 3.56e9·12-s − 3.79e11·13-s + 1.91e12·14-s − 4.52e12·16-s + 1.18e13·17-s + 5.12e12·18-s − 3.41e12·19-s − 7.68e13·21-s + 2.09e14·22-s + 2.05e14·23-s + 1.76e14·24-s − 5.57e14·26-s − 2.05e14·27-s + 7.86e13·28-s + 1.97e15·29-s − 1.01e15·31-s − 3.66e14·32-s − 8.41e15·33-s + 1.74e16·34-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.577·3-s + 0.0288·4-s − 0.585·6-s + 1.74·7-s − 0.985·8-s + 0.333·9-s + 1.65·11-s − 0.0166·12-s − 0.763·13-s + 1.76·14-s − 1.02·16-s + 1.43·17-s + 0.338·18-s − 0.127·19-s − 1.00·21-s + 1.68·22-s + 1.03·23-s + 0.568·24-s − 0.774·26-s − 0.192·27-s + 0.0501·28-s + 0.872·29-s − 0.223·31-s − 0.0576·32-s − 0.956·33-s + 1.45·34-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
| \(L(11)\) |
\(\approx\) |
\(4.300520457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.300520457\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 1.46e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 1.30e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.42e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 3.79e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.18e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.41e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.05e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.97e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.01e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 6.16e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.68e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.10e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.10e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 6.01e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 6.49e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.32e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.32e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.69e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 7.07e18T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.25e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.70e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.11e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.04e20T + 5.27e41T^{2} \) |
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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
−11.07479417682503191934537076059, −9.609662184397930068926656392980, −8.470067678129109333940398174533, −7.19241738563064429018943095094, −5.97320469529129611695977682538, −4.96876074951980320799540968298, −4.45396720765873650147973112615, −3.26557275693413263703657136800, −1.68614128858409986911934763129, −0.824328911787107713532907486733,
0.824328911787107713532907486733, 1.68614128858409986911934763129, 3.26557275693413263703657136800, 4.45396720765873650147973112615, 4.96876074951980320799540968298, 5.97320469529129611695977682538, 7.19241738563064429018943095094, 8.470067678129109333940398174533, 9.609662184397930068926656392980, 11.07479417682503191934537076059
