| L(s) = 1 | − 16·2-s + 81·3-s + 256·4-s − 1.29e3·6-s + 1.03e4·7-s − 4.09e3·8-s + 6.56e3·9-s + 2.74e4·11-s + 2.07e4·12-s + 1.69e5·13-s − 1.65e5·14-s + 6.55e4·16-s + 3.85e5·17-s − 1.04e5·18-s − 6.37e5·19-s + 8.37e5·21-s − 4.38e5·22-s + 1.29e6·23-s − 3.31e5·24-s − 2.71e6·26-s + 5.31e5·27-s + 2.64e6·28-s + 7.16e6·29-s − 7.03e6·31-s − 1.04e6·32-s + 2.22e6·33-s − 6.16e6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.62·7-s − 0.353·8-s + 1/3·9-s + 0.564·11-s + 0.288·12-s + 1.64·13-s − 1.15·14-s + 1/4·16-s + 1.11·17-s − 0.235·18-s − 1.12·19-s + 0.939·21-s − 0.399·22-s + 0.967·23-s − 0.204·24-s − 1.16·26-s + 0.192·27-s + 0.813·28-s + 1.88·29-s − 1.36·31-s − 0.176·32-s + 0.326·33-s − 0.790·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.973611373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.973611373\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10336 T + p^{9} T^{2} \) |
| 11 | \( 1 - 27420 T + p^{9} T^{2} \) |
| 13 | \( 1 - 169762 T + p^{9} T^{2} \) |
| 17 | \( 1 - 385086 T + p^{9} T^{2} \) |
| 19 | \( 1 + 637084 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1298400 T + p^{9} T^{2} \) |
| 29 | \( 1 - 7162974 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7031872 T + p^{9} T^{2} \) |
| 37 | \( 1 + 1926038 T + p^{9} T^{2} \) |
| 41 | \( 1 - 8896074 T + p^{9} T^{2} \) |
| 43 | \( 1 + 32429444 T + p^{9} T^{2} \) |
| 47 | \( 1 + 17206440 T + p^{9} T^{2} \) |
| 53 | \( 1 - 20642154 T + p^{9} T^{2} \) |
| 59 | \( 1 + 63193380 T + p^{9} T^{2} \) |
| 61 | \( 1 + 63758050 T + p^{9} T^{2} \) |
| 67 | \( 1 + 145261964 T + p^{9} T^{2} \) |
| 71 | \( 1 + 367656840 T + p^{9} T^{2} \) |
| 73 | \( 1 + 252486218 T + p^{9} T^{2} \) |
| 79 | \( 1 + 185523712 T + p^{9} T^{2} \) |
| 83 | \( 1 - 467897652 T + p^{9} T^{2} \) |
| 89 | \( 1 - 579096378 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1314516862 T + p^{9} T^{2} \) |
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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
−11.10457267144692273995282509198, −10.36410376204140564266169489302, −8.831347091068309136721117965003, −8.460558735155254983537698010923, −7.43923620897272083938205794571, −6.13091711451236031409059843808, −4.65249425544413666262008889957, −3.33024373369286227056082531909, −1.74156664477443984896677824251, −1.08989811744902611264304960462,
1.08989811744902611264304960462, 1.74156664477443984896677824251, 3.33024373369286227056082531909, 4.65249425544413666262008889957, 6.13091711451236031409059843808, 7.43923620897272083938205794571, 8.460558735155254983537698010923, 8.831347091068309136721117965003, 10.36410376204140564266169489302, 11.10457267144692273995282509198
