| L(s) = 1 | − 2.54·2-s − 1.71·3-s + 4.49·4-s + 5-s + 4.37·6-s − 3.27·7-s − 6.36·8-s − 0.0475·9-s − 2.54·10-s − 7.72·12-s + 13-s + 8.35·14-s − 1.71·15-s + 7.22·16-s − 7.30·17-s + 0.121·18-s + 2.61·19-s + 4.49·20-s + 5.63·21-s − 8.26·23-s + 10.9·24-s + 25-s − 2.54·26-s + 5.23·27-s − 14.7·28-s − 7.56·29-s + 4.37·30-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.992·3-s + 2.24·4-s + 0.447·5-s + 1.78·6-s − 1.23·7-s − 2.25·8-s − 0.0158·9-s − 0.806·10-s − 2.23·12-s + 0.277·13-s + 2.23·14-s − 0.443·15-s + 1.80·16-s − 1.77·17-s + 0.0285·18-s + 0.600·19-s + 1.00·20-s + 1.22·21-s − 1.72·23-s + 2.23·24-s + 0.200·25-s − 0.499·26-s + 1.00·27-s − 2.78·28-s − 1.40·29-s + 0.799·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01283813227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01283813227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 1.71T + 3T^{2} \) |
| 7 | \( 1 + 3.27T + 7T^{2} \) |
| 17 | \( 1 + 7.30T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 8.26T + 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 - 7.69T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 - 3.57T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 + 9.18T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 5.91T + 89T^{2} \) |
| 97 | \( 1 + 2.19T + 97T^{2} \) |
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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
−7.923195227936012906734297483826, −7.13015454974244827040419408969, −6.48454968601704790053923394642, −6.15908193399622083583849449794, −5.51080376940651431028993013855, −4.29730088190586068138643693167, −3.17421119714605115757422372466, −2.31479841986336520687525316360, −1.46345785829413582558043112094, −0.07790808363368444708910096297,
0.07790808363368444708910096297, 1.46345785829413582558043112094, 2.31479841986336520687525316360, 3.17421119714605115757422372466, 4.29730088190586068138643693167, 5.51080376940651431028993013855, 6.15908193399622083583849449794, 6.48454968601704790053923394642, 7.13015454974244827040419408969, 7.923195227936012906734297483826
