| L(s) = 1 | + 39.3·2-s − 81·3-s + 1.03e3·4-s + 1.74e3·5-s − 3.18e3·6-s + 7.50e3·7-s + 2.07e4·8-s + 6.56e3·9-s + 6.86e4·10-s + 2.27e3·11-s − 8.41e4·12-s + 8.93e4·13-s + 2.95e5·14-s − 1.41e5·15-s + 2.85e5·16-s + 5.35e5·17-s + 2.58e5·18-s − 1.00e6·19-s + 1.81e6·20-s − 6.07e5·21-s + 8.94e4·22-s + 1.67e6·23-s − 1.68e6·24-s + 1.08e6·25-s + 3.51e6·26-s − 5.31e5·27-s + 7.79e6·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.577·3-s + 2.02·4-s + 1.24·5-s − 1.00·6-s + 1.18·7-s + 1.79·8-s + 0.333·9-s + 2.17·10-s + 0.0467·11-s − 1.17·12-s + 0.867·13-s + 2.05·14-s − 0.720·15-s + 1.08·16-s + 1.55·17-s + 0.580·18-s − 1.77·19-s + 2.53·20-s − 0.681·21-s + 0.0813·22-s + 1.24·23-s − 1.03·24-s + 0.556·25-s + 1.50·26-s − 0.192·27-s + 2.39·28-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
| \(L(5)\) |
\(\approx\) |
\(8.648132438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.648132438\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 39.3T + 512T^{2} \) |
| 5 | \( 1 - 1.74e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.50e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.27e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.35e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.00e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.67e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.85e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.35e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.61e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.89e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.72e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.66e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.77e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.50e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.47e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.53e8T + 7.60e17T^{2} \) |
| 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
−10.98430486438609669162575677273, −10.75752001466425430623444481850, −9.070336436397461811331415816402, −7.48591136027160224629235607239, −6.22713406348288973265078200273, −5.61415931069208026049732252924, −4.84558479469383373270395734506, −3.67473592596106847721146062580, −2.17305024247224151534387906841, −1.34261806809496649004310475679,
1.34261806809496649004310475679, 2.17305024247224151534387906841, 3.67473592596106847721146062580, 4.84558479469383373270395734506, 5.61415931069208026049732252924, 6.22713406348288973265078200273, 7.48591136027160224629235607239, 9.070336436397461811331415816402, 10.75752001466425430623444481850, 10.98430486438609669162575677273
