| L(s) = 1 | + 22.5·2-s − 238.·3-s − 1.54e3·4-s + 6.83e3·5-s − 5.37e3·6-s − 6.22e4·7-s − 8.07e4·8-s − 1.20e5·9-s + 1.53e5·10-s − 1.40e5·11-s + 3.67e5·12-s − 1.40e6·13-s − 1.40e6·14-s − 1.63e6·15-s + 1.33e6·16-s − 7.22e5·17-s − 2.70e6·18-s − 5.33e6·19-s − 1.05e7·20-s + 1.48e7·21-s − 3.16e6·22-s − 1.14e7·23-s + 1.92e7·24-s − 2.05e6·25-s − 3.17e7·26-s + 7.09e7·27-s + 9.59e7·28-s + ⋯ |
| L(s) = 1 | + 0.497·2-s − 0.567·3-s − 0.752·4-s + 0.978·5-s − 0.282·6-s − 1.39·7-s − 0.871·8-s − 0.678·9-s + 0.486·10-s − 0.263·11-s + 0.426·12-s − 1.05·13-s − 0.695·14-s − 0.555·15-s + 0.319·16-s − 0.123·17-s − 0.337·18-s − 0.493·19-s − 0.736·20-s + 0.793·21-s − 0.131·22-s − 0.372·23-s + 0.494·24-s − 0.0420·25-s − 0.523·26-s + 0.951·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4883637281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4883637281\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + 1.15e10T \) |
| good | 2 | \( 1 - 22.5T + 2.04e3T^{2} \) |
| 3 | \( 1 + 238.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 6.83e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.22e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.40e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.40e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.22e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 5.33e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.14e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.30e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 5.67e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.41e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 4.61e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.46e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.19e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.62e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.58e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.20e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.00e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.98e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 9.21e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.11e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.29e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.25e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.31e10T + 7.15e21T^{2} \) |
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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
−11.96664140039845845079574735446, −10.31681534397552446982422265010, −9.651940177622077381518236015706, −8.643758469003914197204601502873, −6.76030928121165178084191950085, −5.81485014848979691854527724468, −5.09141985553782854291518118949, −3.54617628269503133181194071955, −2.37019813387952516793329013001, −0.31770133880905805292140603485,
0.31770133880905805292140603485, 2.37019813387952516793329013001, 3.54617628269503133181194071955, 5.09141985553782854291518118949, 5.81485014848979691854527724468, 6.76030928121165178084191950085, 8.643758469003914197204601502873, 9.651940177622077381518236015706, 10.31681534397552446982422265010, 11.96664140039845845079574735446
