| L(s) = 1 | + 1.71e3·2-s + 5.90e4·3-s + 8.50e5·4-s + 1.01e8·6-s + 6.24e8·7-s − 2.14e9·8-s + 3.48e9·9-s + 1.47e11·11-s + 5.02e10·12-s + 5.50e11·13-s + 1.07e12·14-s − 5.45e12·16-s + 4.04e12·17-s + 5.98e12·18-s + 1.22e12·19-s + 3.68e13·21-s + 2.53e14·22-s + 2.50e14·23-s − 1.26e14·24-s + 9.45e14·26-s + 2.05e14·27-s + 5.31e14·28-s − 3.49e15·29-s − 2.93e15·31-s − 4.88e15·32-s + 8.70e15·33-s + 6.95e15·34-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.577·3-s + 0.405·4-s + 0.684·6-s + 0.835·7-s − 0.704·8-s + 0.333·9-s + 1.71·11-s + 0.234·12-s + 1.10·13-s + 0.990·14-s − 1.24·16-s + 0.487·17-s + 0.395·18-s + 0.0459·19-s + 0.482·21-s + 2.03·22-s + 1.26·23-s − 0.406·24-s + 1.31·26-s + 0.192·27-s + 0.338·28-s − 1.54·29-s − 0.642·31-s − 0.766·32-s + 0.989·33-s + 0.577·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(7.302515094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.302515094\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.71e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 6.24e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.47e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.50e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 4.04e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.22e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.50e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.49e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.93e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.06e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.45e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.05e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.95e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.24e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 7.20e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 4.77e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 7.70e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.63e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.29e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 9.55e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.20e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.15e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 2.46e20T + 5.27e41T^{2} \) |
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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
−11.03264334981024580418033484094, −9.272339980075480112314865436984, −8.681829769185694943681039639907, −7.22639530576322043111833302287, −6.10036355139680912497850044115, −5.03634442319298137205168198625, −3.90478760883498069022513968770, −3.43366841103990700938123090627, −1.92671607588297497464029408750, −0.982133164506860285995281688304,
0.982133164506860285995281688304, 1.92671607588297497464029408750, 3.43366841103990700938123090627, 3.90478760883498069022513968770, 5.03634442319298137205168198625, 6.10036355139680912497850044115, 7.22639530576322043111833302287, 8.681829769185694943681039639907, 9.272339980075480112314865436984, 11.03264334981024580418033484094
