L(s) = 1 | + 1.20·2-s − 27·3-s − 126.·4-s − 262.·5-s − 32.4·6-s + 1.19e3·7-s − 306.·8-s + 729·9-s − 315.·10-s + 3.81e3·11-s + 3.41e3·12-s + 5.19e3·13-s + 1.43e3·14-s + 7.08e3·15-s + 1.58e4·16-s + 3.55e4·17-s + 876.·18-s + 2.80e4·19-s + 3.32e4·20-s − 3.22e4·21-s + 4.58e3·22-s + 5.37e3·23-s + 8.26e3·24-s − 9.18e3·25-s + 6.25e3·26-s − 1.96e4·27-s − 1.51e5·28-s + ⋯ |
L(s) = 1 | + 0.106·2-s − 0.577·3-s − 0.988·4-s − 0.939·5-s − 0.0613·6-s + 1.31·7-s − 0.211·8-s + 0.333·9-s − 0.0998·10-s + 0.863·11-s + 0.570·12-s + 0.656·13-s + 0.140·14-s + 0.542·15-s + 0.966·16-s + 1.75·17-s + 0.0354·18-s + 0.937·19-s + 0.928·20-s − 0.760·21-s + 0.0917·22-s + 0.0920·23-s + 0.122·24-s − 0.117·25-s + 0.0697·26-s − 0.192·27-s − 1.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.830459474\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.830459474\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
good | 2 | \( 1 - 1.20T + 128T^{2} \) |
| 5 | \( 1 + 262.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.19e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.19e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.55e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.80e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.37e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.10e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.32e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.34e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.45e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.61e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.47e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.03e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.96e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.58e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.77e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.28e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.29e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.00e7T + 8.07e13T^{2} \) |
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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
−9.860266087344746481253986774095, −8.887887806305073643288667208110, −7.977733129134880015878680883602, −7.41990151783753233864737875078, −5.83734592526124149905722839189, −5.13101069957941307924785291092, −4.14199922173602065440209615888, −3.50441223032403512700541344624, −1.38013203065727561657756961058, −0.72184385706028779940259732636,
0.72184385706028779940259732636, 1.38013203065727561657756961058, 3.50441223032403512700541344624, 4.14199922173602065440209615888, 5.13101069957941307924785291092, 5.83734592526124149905722839189, 7.41990151783753233864737875078, 7.977733129134880015878680883602, 8.887887806305073643288667208110, 9.860266087344746481253986774095