| L(s) = 1 | + 19.6·2-s − 27·3-s + 258.·4-s − 436.·5-s − 530.·6-s + 956.·7-s + 2.56e3·8-s + 729·9-s − 8.58e3·10-s + 1.97e3·11-s − 6.97e3·12-s − 5.30e3·13-s + 1.88e4·14-s + 1.17e4·15-s + 1.72e4·16-s − 1.91e4·17-s + 1.43e4·18-s − 2.14e4·19-s − 1.12e5·20-s − 2.58e4·21-s + 3.87e4·22-s − 3.43e4·23-s − 6.91e4·24-s + 1.12e5·25-s − 1.04e5·26-s − 1.96e4·27-s + 2.47e5·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 2.01·4-s − 1.56·5-s − 1.00·6-s + 1.05·7-s + 1.76·8-s + 0.333·9-s − 2.71·10-s + 0.446·11-s − 1.16·12-s − 0.670·13-s + 1.83·14-s + 0.902·15-s + 1.05·16-s − 0.947·17-s + 0.579·18-s − 0.717·19-s − 3.15·20-s − 0.608·21-s + 0.776·22-s − 0.588·23-s − 1.02·24-s + 1.44·25-s − 1.16·26-s − 0.192·27-s + 2.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 19.6T + 128T^{2} \) |
| 5 | \( 1 + 436.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.30e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.91e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.00e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.54e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.69e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.40e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.80e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.27e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.30e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.05e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.36e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.49e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.45e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.41e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.49196865990953424151177364460, −10.61186261529811497687923605761, −8.492516560015961289087758517129, −7.37364678210501767160905896615, −6.51274905947508546758278277448, −5.04918757563897706118054794272, −4.45694433219675826927404614465, −3.57323160214835986380401936263, −1.93410422473796113134792743784, 0,
1.93410422473796113134792743784, 3.57323160214835986380401936263, 4.45694433219675826927404614465, 5.04918757563897706118054794272, 6.51274905947508546758278277448, 7.37364678210501767160905896615, 8.492516560015961289087758517129, 10.61186261529811497687923605761, 11.49196865990953424151177364460
