| L(s) = 1 | − 15.8·2-s − 48.4·3-s − 260.·4-s + 2.42e3·5-s + 769.·6-s − 4.46e3·7-s + 1.22e4·8-s − 1.73e4·9-s − 3.85e4·10-s + 2.19e4·11-s + 1.26e4·12-s + 1.43e5·13-s + 7.08e4·14-s − 1.17e5·15-s − 6.13e4·16-s + 2.75e5·18-s − 8.90e5·19-s − 6.31e5·20-s + 2.16e5·21-s − 3.48e5·22-s + 6.62e5·23-s − 5.94e5·24-s + 3.94e6·25-s − 2.27e6·26-s + 1.79e6·27-s + 1.16e6·28-s − 7.32e6·29-s + ⋯ |
| L(s) = 1 | − 0.701·2-s − 0.345·3-s − 0.507·4-s + 1.73·5-s + 0.242·6-s − 0.703·7-s + 1.05·8-s − 0.880·9-s − 1.21·10-s + 0.452·11-s + 0.175·12-s + 1.39·13-s + 0.493·14-s − 0.600·15-s − 0.234·16-s + 0.617·18-s − 1.56·19-s − 0.882·20-s + 0.242·21-s − 0.317·22-s + 0.493·23-s − 0.365·24-s + 2.01·25-s − 0.975·26-s + 0.649·27-s + 0.357·28-s − 1.92·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 15.8T + 512T^{2} \) |
| 3 | \( 1 + 48.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.42e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.46e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.43e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.90e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.62e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.32e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.46e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.16e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.56e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.26e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.20e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.00e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.48e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.12e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.82e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.49e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.35e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.29e9T + 7.60e17T^{2} \) |
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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
−9.511471185833838407824619156618, −9.101444094580808351492706952434, −8.217670666052634002730366354529, −6.46295358368788484593008347224, −6.08289973356855036325390809801, −4.99515659357575909072534623611, −3.55741833001007376270818328520, −2.12093537902153574039763033078, −1.14868804271285429002098073099, 0,
1.14868804271285429002098073099, 2.12093537902153574039763033078, 3.55741833001007376270818328520, 4.99515659357575909072534623611, 6.08289973356855036325390809801, 6.46295358368788484593008347224, 8.217670666052634002730366354529, 9.101444094580808351492706952434, 9.511471185833838407824619156618
