| L(s) = 1 | − 5.76·2-s − 27·3-s − 94.7·4-s − 461.·5-s + 155.·6-s + 625.·7-s + 1.28e3·8-s + 729·9-s + 2.66e3·10-s + 2.55e3·12-s + 7.05e3·13-s − 3.60e3·14-s + 1.24e4·15-s + 4.71e3·16-s − 2.99e4·17-s − 4.20e3·18-s − 1.90e4·19-s + 4.37e4·20-s − 1.68e4·21-s + 1.07e5·23-s − 3.46e4·24-s + 1.35e5·25-s − 4.06e4·26-s − 1.96e4·27-s − 5.92e4·28-s + 1.42e5·29-s − 7.19e4·30-s + ⋯ |
| L(s) = 1 | − 0.509·2-s − 0.577·3-s − 0.740·4-s − 1.65·5-s + 0.294·6-s + 0.689·7-s + 0.887·8-s + 0.333·9-s + 0.842·10-s + 0.427·12-s + 0.890·13-s − 0.351·14-s + 0.954·15-s + 0.287·16-s − 1.48·17-s − 0.169·18-s − 0.636·19-s + 1.22·20-s − 0.398·21-s + 1.84·23-s − 0.512·24-s + 1.73·25-s − 0.453·26-s − 0.192·27-s − 0.510·28-s + 1.08·29-s − 0.486·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4050091738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4050091738\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 5.76T + 128T^{2} \) |
| 5 | \( 1 + 461.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 625.T + 8.23e5T^{2} \) |
| 13 | \( 1 - 7.05e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.99e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.90e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.07e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.42e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.13e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.03e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.85e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.19e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.24e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.69e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.71e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.17e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.31e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.38e7T + 8.07e13T^{2} \) |
| 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
−10.62023644329985532143152143471, −8.829331278957281435930338048477, −8.639411487517489352678015418783, −7.54335475568113049985203238764, −6.71548099769716377498977673316, −5.02963641135617163007034604699, −4.44310896298515312731446205409, −3.49146173795058303479153105456, −1.45878138390266339021919172951, −0.36289170977770840943162597436,
0.36289170977770840943162597436, 1.45878138390266339021919172951, 3.49146173795058303479153105456, 4.44310896298515312731446205409, 5.02963641135617163007034604699, 6.71548099769716377498977673316, 7.54335475568113049985203238764, 8.639411487517489352678015418783, 8.829331278957281435930338048477, 10.62023644329985532143152143471
