| L(s) = 1 | − 32·2-s + 1.02e3·4-s − 4.33e3·5-s − 1.68e4·7-s − 3.27e4·8-s + 1.38e5·10-s + 8.97e5·11-s + 1.39e6·13-s + 5.37e5·14-s + 1.04e6·16-s − 6.38e6·17-s − 1.07e7·19-s − 4.43e6·20-s − 2.87e7·22-s + 1.98e7·23-s − 3.00e7·25-s − 4.47e7·26-s − 1.72e7·28-s + 1.59e8·29-s + 9.87e7·31-s − 3.35e7·32-s + 2.04e8·34-s + 7.28e7·35-s − 5.14e8·37-s + 3.44e8·38-s + 1.42e8·40-s − 3.05e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.620·5-s − 0.377·7-s − 0.353·8-s + 0.438·10-s + 1.68·11-s + 1.04·13-s + 0.267·14-s + 0.250·16-s − 1.09·17-s − 0.997·19-s − 0.310·20-s − 1.18·22-s + 0.642·23-s − 0.615·25-s − 0.739·26-s − 0.188·28-s + 1.44·29-s + 0.619·31-s − 0.176·32-s + 0.771·34-s + 0.234·35-s − 1.21·37-s + 0.705·38-s + 0.219·40-s − 0.411·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.252388338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252388338\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 + 4.33e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 8.97e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.39e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.38e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.07e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.98e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.59e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 9.87e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.14e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.05e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.84e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 4.60e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.01e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.53e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.16e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.14e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.09e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.15e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.58e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.49e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.92e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.12e11T + 7.15e21T^{2} \) |
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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
−11.27717081228115862424003776223, −10.17853544843772225359941504053, −8.920212075909347251467162914746, −8.422393682029051119824719678111, −6.84857298227846274928750107060, −6.32074738623030359969391672394, −4.37061726311795727696209845282, −3.36749004923450608766431976523, −1.76707190194669353040239185770, −0.61860396492550703944881395599,
0.61860396492550703944881395599, 1.76707190194669353040239185770, 3.36749004923450608766431976523, 4.37061726311795727696209845282, 6.32074738623030359969391672394, 6.84857298227846274928750107060, 8.422393682029051119824719678111, 8.920212075909347251467162914746, 10.17853544843772225359941504053, 11.27717081228115862424003776223
