| L(s) = 1 | + 42.2·2-s + 81·3-s + 1.27e3·4-s − 60.0·5-s + 3.42e3·6-s + 8.82e3·7-s + 3.22e4·8-s + 6.56e3·9-s − 2.53e3·10-s + 1.62e4·11-s + 1.03e5·12-s + 1.23e5·13-s + 3.73e5·14-s − 4.86e3·15-s + 7.10e5·16-s − 1.65e5·17-s + 2.77e5·18-s − 6.92e5·19-s − 7.65e4·20-s + 7.14e5·21-s + 6.84e5·22-s − 3.87e5·23-s + 2.61e6·24-s − 1.94e6·25-s + 5.21e6·26-s + 5.31e5·27-s + 1.12e7·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.577·3-s + 2.48·4-s − 0.0429·5-s + 1.07·6-s + 1.38·7-s + 2.78·8-s + 0.333·9-s − 0.0802·10-s + 0.333·11-s + 1.43·12-s + 1.19·13-s + 2.59·14-s − 0.0247·15-s + 2.70·16-s − 0.481·17-s + 0.622·18-s − 1.21·19-s − 0.106·20-s + 0.802·21-s + 0.623·22-s − 0.288·23-s + 1.60·24-s − 0.998·25-s + 2.23·26-s + 0.192·27-s + 3.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(11.60042244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.60042244\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 42.2T + 512T^{2} \) |
| 5 | \( 1 + 60.0T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.82e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.62e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.23e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.65e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.92e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.87e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.66e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.05e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.42e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.48e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.30e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.22e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.05e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 5.78e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.31e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.74e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.84e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.49e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.64e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.37e9T + 7.60e17T^{2} \) |
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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
−11.31829156753377835342469981424, −10.58652019674453704678117475511, −8.703133831659072834656613766027, −7.73542509393641340054247013667, −6.54701039956571401456590672372, −5.53090460029737518370946168683, −4.34375572180625058566327585038, −3.78672741388179527864140700066, −2.30839817299573472746727752598, −1.52886575134937182375786154549,
1.52886575134937182375786154549, 2.30839817299573472746727752598, 3.78672741388179527864140700066, 4.34375572180625058566327585038, 5.53090460029737518370946168683, 6.54701039956571401456590672372, 7.73542509393641340054247013667, 8.703133831659072834656613766027, 10.58652019674453704678117475511, 11.31829156753377835342469981424
