| L(s) = 1 | + 16·2-s + 170·3-s + 256·4-s + 544·5-s + 2.72e3·6-s − 2.40e3·7-s + 4.09e3·8-s + 9.21e3·9-s + 8.70e3·10-s + 4.88e4·11-s + 4.35e4·12-s − 1.58e4·13-s − 3.84e4·14-s + 9.24e4·15-s + 6.55e4·16-s − 2.14e4·17-s + 1.47e5·18-s − 7.16e5·19-s + 1.39e5·20-s − 4.08e5·21-s + 7.81e5·22-s − 2.47e6·23-s + 6.96e5·24-s − 1.65e6·25-s − 2.54e5·26-s − 1.77e6·27-s − 6.14e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.21·3-s + 1/2·4-s + 0.389·5-s + 0.856·6-s − 0.377·7-s + 0.353·8-s + 0.468·9-s + 0.275·10-s + 1.00·11-s + 0.605·12-s − 0.154·13-s − 0.267·14-s + 0.471·15-s + 1/4·16-s − 0.0621·17-s + 0.331·18-s − 1.26·19-s + 0.194·20-s − 0.457·21-s + 0.710·22-s − 1.84·23-s + 0.428·24-s − 0.848·25-s − 0.109·26-s − 0.644·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.447985088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447985088\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 7 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 - 170 T + p^{9} T^{2} \) |
| 5 | \( 1 - 544 T + p^{9} T^{2} \) |
| 11 | \( 1 - 48824 T + p^{9} T^{2} \) |
| 13 | \( 1 + 15876 T + p^{9} T^{2} \) |
| 17 | \( 1 + 21418 T + p^{9} T^{2} \) |
| 19 | \( 1 + 716410 T + p^{9} T^{2} \) |
| 23 | \( 1 + 2470000 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5556826 T + p^{9} T^{2} \) |
| 31 | \( 1 - 5799348 T + p^{9} T^{2} \) |
| 37 | \( 1 + 3894430 T + p^{9} T^{2} \) |
| 41 | \( 1 + 6360858 T + p^{9} T^{2} \) |
| 43 | \( 1 + 18701296 T + p^{9} T^{2} \) |
| 47 | \( 1 - 56539068 T + p^{9} T^{2} \) |
| 53 | \( 1 + 59894682 T + p^{9} T^{2} \) |
| 59 | \( 1 - 165629662 T + p^{9} T^{2} \) |
| 61 | \( 1 - 51419016 T + p^{9} T^{2} \) |
| 67 | \( 1 - 93546508 T + p^{9} T^{2} \) |
| 71 | \( 1 + 95633536 T + p^{9} T^{2} \) |
| 73 | \( 1 - 306496402 T + p^{9} T^{2} \) |
| 79 | \( 1 - 496474152 T + p^{9} T^{2} \) |
| 83 | \( 1 + 371486962 T + p^{9} T^{2} \) |
| 89 | \( 1 + 165482550 T + p^{9} T^{2} \) |
| 97 | \( 1 - 758016742 T + p^{9} T^{2} \) |
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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
−17.29329635756774406635425818483, −15.64484234561526554061895252465, −14.37424775534783358622940760164, −13.62270424386991541147928370074, −12.07239598431441271241500168481, −9.938313715044090534754168608426, −8.368249670529536252547517682803, −6.37302806691458720870578257616, −3.93232421548929015444637441295, −2.23570180506101370749220624969,
2.23570180506101370749220624969, 3.93232421548929015444637441295, 6.37302806691458720870578257616, 8.368249670529536252547517682803, 9.938313715044090534754168608426, 12.07239598431441271241500168481, 13.62270424386991541147928370074, 14.37424775534783358622940760164, 15.64484234561526554061895252465, 17.29329635756774406635425818483
