| L(s) = 1 | + 18.0·2-s − 243·3-s − 1.72e3·4-s − 3.87e3·5-s − 4.37e3·6-s + 3.21e4·7-s − 6.79e4·8-s + 5.90e4·9-s − 6.99e4·10-s − 3.62e5·11-s + 4.18e5·12-s − 2.86e5·13-s + 5.78e5·14-s + 9.42e5·15-s + 2.30e6·16-s + 9.02e6·17-s + 1.06e6·18-s − 2.13e7·19-s + 6.68e6·20-s − 7.80e6·21-s − 6.53e6·22-s − 5.38e7·23-s + 1.65e7·24-s − 3.37e7·25-s − 5.15e6·26-s − 1.43e7·27-s − 5.53e7·28-s + ⋯ |
| L(s) = 1 | + 0.398·2-s − 0.577·3-s − 0.841·4-s − 0.555·5-s − 0.229·6-s + 0.722·7-s − 0.733·8-s + 0.333·9-s − 0.221·10-s − 0.678·11-s + 0.485·12-s − 0.213·13-s + 0.287·14-s + 0.320·15-s + 0.549·16-s + 1.54·17-s + 0.132·18-s − 1.97·19-s + 0.467·20-s − 0.417·21-s − 0.270·22-s − 1.74·23-s + 0.423·24-s − 0.691·25-s − 0.0850·26-s − 0.192·27-s − 0.607·28-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
| \(L(6)\) |
\(\approx\) |
\(0.5752139530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5752139530\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 18.0T + 2.04e3T^{2} \) |
| 5 | \( 1 + 3.87e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.21e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.62e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.86e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 9.02e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.13e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.38e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.58e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.44e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.68e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.21e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.48e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.87e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.62e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 3.26e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.11e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.81e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 4.66e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.85e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.48e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.91e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.14e11T + 7.15e21T^{2} \) |
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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
−10.62536011203534995329977568880, −9.892133426687273394130672646578, −8.361339336372514842387511474944, −7.88566220539495974313901114808, −6.28180236690412689389904387822, −5.26032613973471741766316675967, −4.46163687346093482783285757083, −3.49937964376422357827764273925, −1.82782535021431123555295531877, −0.32958503739156362304835166218,
0.32958503739156362304835166218, 1.82782535021431123555295531877, 3.49937964376422357827764273925, 4.46163687346093482783285757083, 5.26032613973471741766316675967, 6.28180236690412689389904387822, 7.88566220539495974313901114808, 8.361339336372514842387511474944, 9.892133426687273394130672646578, 10.62536011203534995329977568880
