| L(s) = 1 | − 4.20·2-s − 17.5·3-s − 14.3·4-s + 73.9·6-s + 253.·7-s + 194.·8-s + 65.7·9-s − 360.·11-s + 251.·12-s + 40.7·13-s − 1.06e3·14-s − 361.·16-s − 218.·17-s − 276.·18-s − 807.·19-s − 4.45e3·21-s + 1.51e3·22-s + 4.17e3·23-s − 3.42e3·24-s − 171.·26-s + 3.11e3·27-s − 3.63e3·28-s + 8.77e3·29-s + 2.57e3·31-s − 4.71e3·32-s + 6.34e3·33-s + 920.·34-s + ⋯ |
| L(s) = 1 | − 0.743·2-s − 1.12·3-s − 0.447·4-s + 0.838·6-s + 1.95·7-s + 1.07·8-s + 0.270·9-s − 0.899·11-s + 0.504·12-s + 0.0668·13-s − 1.45·14-s − 0.352·16-s − 0.183·17-s − 0.201·18-s − 0.513·19-s − 2.20·21-s + 0.668·22-s + 1.64·23-s − 1.21·24-s − 0.0496·26-s + 0.822·27-s − 0.875·28-s + 1.93·29-s + 0.481·31-s − 0.813·32-s + 1.01·33-s + 0.136·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.083429487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083429487\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 4.20T + 32T^{2} \) |
| 3 | \( 1 + 17.5T + 243T^{2} \) |
| 7 | \( 1 - 253.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 360.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 40.7T + 3.71e5T^{2} \) |
| 17 | \( 1 + 218.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 807.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.49e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.13e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.43e5T + 8.58e9T^{2} \) |
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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
−8.957989258524677641298182873966, −8.337330466975555083220435602020, −7.75651835478058608430087335504, −6.74585151665260659379579388512, −5.51173149831647486199288487879, −4.86416326938495493962987786255, −4.45547727839357815214758867159, −2.54855262657065910026268152641, −1.24850639398037044778601132906, −0.63881965402925697489927065579,
0.63881965402925697489927065579, 1.24850639398037044778601132906, 2.54855262657065910026268152641, 4.45547727839357815214758867159, 4.86416326938495493962987786255, 5.51173149831647486199288487879, 6.74585151665260659379579388512, 7.75651835478058608430087335504, 8.337330466975555083220435602020, 8.957989258524677641298182873966
