| L(s) = 1 | − 31.4·2-s + 60.4·3-s + 474.·4-s − 1.58e3·5-s − 1.89e3·6-s + 1.04e4·7-s + 1.16e3·8-s − 1.60e4·9-s + 4.97e4·10-s + 7.81e4·11-s + 2.86e4·12-s + 1.11e5·13-s − 3.29e5·14-s − 9.56e4·15-s − 2.79e5·16-s + 5.03e5·18-s − 2.02e5·19-s − 7.51e5·20-s + 6.33e5·21-s − 2.45e6·22-s − 7.59e5·23-s + 7.03e4·24-s + 5.50e5·25-s − 3.50e6·26-s − 2.15e6·27-s + 4.98e6·28-s − 7.48e6·29-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.430·3-s + 0.927·4-s − 1.13·5-s − 0.597·6-s + 1.65·7-s + 0.100·8-s − 0.814·9-s + 1.57·10-s + 1.61·11-s + 0.399·12-s + 1.08·13-s − 2.29·14-s − 0.487·15-s − 1.06·16-s + 1.13·18-s − 0.356·19-s − 1.05·20-s + 0.711·21-s − 2.23·22-s − 0.565·23-s + 0.0432·24-s + 0.281·25-s − 1.50·26-s − 0.781·27-s + 1.53·28-s − 1.96·29-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 | \( 1 + 31.4T + 512T^{2} \) |
| 3 | \( 1 - 60.4T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.58e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.04e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.81e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.11e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 2.02e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.59e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.48e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.06e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.12e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.31e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.13e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.33e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.46e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.34e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.24e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.42e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.13e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.26e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.37e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.60e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.52e9T + 7.60e17T^{2} \) |
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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
−9.318487016360240865075148157945, −8.718658230756937621332321265227, −8.034619290034863248504070768374, −7.52297218230729775057001932777, −6.10656322692609609938405326921, −4.46183064213055367265121502383, −3.67283905633517975045811132270, −1.91454598057749625918757303751, −1.18352985641386299158686833167, 0,
1.18352985641386299158686833167, 1.91454598057749625918757303751, 3.67283905633517975045811132270, 4.46183064213055367265121502383, 6.10656322692609609938405326921, 7.52297218230729775057001932777, 8.034619290034863248504070768374, 8.718658230756937621332321265227, 9.318487016360240865075148157945
