| L(s) = 1 | − 13.0·2-s + 41.0·4-s + 125·5-s − 107.·7-s + 1.13e3·8-s − 1.62e3·10-s − 2.56e3·11-s − 1.45e4·13-s + 1.40e3·14-s − 1.99e4·16-s + 1.30e4·17-s − 2.07e4·19-s + 5.13e3·20-s + 3.33e4·22-s − 6.19e4·23-s + 1.56e4·25-s + 1.88e5·26-s − 4.42e3·28-s − 1.85e5·29-s + 2.35e5·31-s + 1.14e5·32-s − 1.69e5·34-s − 1.34e4·35-s + 4.29e5·37-s + 2.69e5·38-s + 1.41e5·40-s + 3.06e5·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.320·4-s + 0.447·5-s − 0.118·7-s + 0.780·8-s − 0.513·10-s − 0.580·11-s − 1.83·13-s + 0.136·14-s − 1.21·16-s + 0.642·17-s − 0.693·19-s + 0.143·20-s + 0.667·22-s − 1.06·23-s + 0.199·25-s + 2.10·26-s − 0.0380·28-s − 1.41·29-s + 1.42·31-s + 0.618·32-s − 0.738·34-s − 0.0531·35-s + 1.39·37-s + 0.797·38-s + 0.349·40-s + 0.693·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5016856136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5016856136\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| good | 2 | \( 1 + 13.0T + 128T^{2} \) |
| 7 | \( 1 + 107.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.45e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.30e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.07e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.85e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.35e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.29e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.06e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.42e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.27e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.02e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.02e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.72e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.81e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.08e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.25e6T + 8.07e13T^{2} \) |
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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
−9.814507269827344916861357957566, −9.483661149609963891039620181206, −8.147493160343104976242475339783, −7.68079905125975689471240171374, −6.56875851162300575494241565101, −5.31048588599468529451681876147, −4.35903322098039183126392956498, −2.67337614441427254763282779362, −1.75418913512080214710005431867, −0.37959165308324565153156785042,
0.37959165308324565153156785042, 1.75418913512080214710005431867, 2.67337614441427254763282779362, 4.35903322098039183126392956498, 5.31048588599468529451681876147, 6.56875851162300575494241565101, 7.68079905125975689471240171374, 8.147493160343104976242475339783, 9.483661149609963891039620181206, 9.814507269827344916861357957566
