| L(s) = 1 | + 23.7·2-s + 243·3-s − 1.48e3·4-s + 6.92e3·5-s + 5.77e3·6-s + 2.44e4·7-s − 8.39e4·8-s + 5.90e4·9-s + 1.64e5·10-s − 8.88e5·11-s − 3.60e5·12-s − 9.14e5·13-s + 5.82e5·14-s + 1.68e6·15-s + 1.04e6·16-s − 3.69e6·17-s + 1.40e6·18-s + 4.55e6·19-s − 1.02e7·20-s + 5.95e6·21-s − 2.11e7·22-s + 7.76e6·23-s − 2.03e7·24-s − 9.16e5·25-s − 2.17e7·26-s + 1.43e7·27-s − 3.63e7·28-s + ⋯ |
| L(s) = 1 | + 0.525·2-s + 0.577·3-s − 0.723·4-s + 0.990·5-s + 0.303·6-s + 0.550·7-s − 0.905·8-s + 0.333·9-s + 0.520·10-s − 1.66·11-s − 0.417·12-s − 0.683·13-s + 0.289·14-s + 0.571·15-s + 0.248·16-s − 0.630·17-s + 0.175·18-s + 0.421·19-s − 0.717·20-s + 0.317·21-s − 0.873·22-s + 0.251·23-s − 0.522·24-s − 0.0187·25-s − 0.358·26-s + 0.192·27-s − 0.398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.252013420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.252013420\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 23.7T + 2.04e3T^{2} \) |
| 5 | \( 1 - 6.92e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.44e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.88e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 9.14e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.69e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.55e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 7.76e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.05e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 3.06e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.90e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.24e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.44e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 4.59e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.43e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.80e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.65e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.44e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.35e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.48e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.34e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.37e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.27e11T + 7.15e21T^{2} \) |
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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
−10.31104830023927051605195024814, −9.786049193682013148213623514545, −8.606650272533057915791973941099, −7.85562679401162438340497638761, −6.34101130842574954872580774148, −5.12096047180796498595361573299, −4.63298574825049523657492782314, −2.97696677991999807281846914568, −2.26318025424946773338884461636, −0.72574860186572760279920359078,
0.72574860186572760279920359078, 2.26318025424946773338884461636, 2.97696677991999807281846914568, 4.63298574825049523657492782314, 5.12096047180796498595361573299, 6.34101130842574954872580774148, 7.85562679401162438340497638761, 8.606650272533057915791973941099, 9.786049193682013148213623514545, 10.31104830023927051605195024814
