| L(s) = 1 | + 14.7·2-s + 89.3·4-s + 125·5-s − 343·7-s − 569.·8-s + 1.84e3·10-s − 91.4·11-s + 2.26e3·13-s − 5.05e3·14-s − 1.98e4·16-s + 2.70e4·17-s + 2.86e4·19-s + 1.11e4·20-s − 1.34e3·22-s − 1.05e5·23-s + 1.56e4·25-s + 3.33e4·26-s − 3.06e4·28-s + 1.44e5·29-s + 2.43e5·31-s − 2.19e5·32-s + 3.99e5·34-s − 4.28e4·35-s − 3.79e5·37-s + 4.21e5·38-s − 7.12e4·40-s + 3.35e5·41-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.697·4-s + 0.447·5-s − 0.377·7-s − 0.393·8-s + 0.582·10-s − 0.0207·11-s + 0.285·13-s − 0.492·14-s − 1.21·16-s + 1.33·17-s + 0.956·19-s + 0.312·20-s − 0.0270·22-s − 1.80·23-s + 0.199·25-s + 0.372·26-s − 0.263·28-s + 1.10·29-s + 1.46·31-s − 1.18·32-s + 1.74·34-s − 0.169·35-s − 1.23·37-s + 1.24·38-s − 0.176·40-s + 0.759·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.628115123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.628115123\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 - 14.7T + 128T^{2} \) |
| 11 | \( 1 + 91.4T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.26e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.05e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.43e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.35e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.24e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.49e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.47e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.70e3T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.42e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.78e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.24e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.12e7T + 8.07e13T^{2} \) |
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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
−10.43254239305531713885002281356, −9.700316098053021543961325793567, −8.553468854885378106399714371407, −7.31007014230634680479851869985, −6.07606182156998626289142226947, −5.60856264868042493587990654961, −4.40378916127145478529069286544, −3.42279480739007068341152528755, −2.45662367712936081203689038223, −0.869089727176070460892081697141,
0.869089727176070460892081697141, 2.45662367712936081203689038223, 3.42279480739007068341152528755, 4.40378916127145478529069286544, 5.60856264868042493587990654961, 6.07606182156998626289142226947, 7.31007014230634680479851869985, 8.553468854885378106399714371407, 9.700316098053021543961325793567, 10.43254239305531713885002281356
