| L(s) = 1 | − 15.9·3-s + 31.5·5-s − 106.·7-s + 11.9·9-s + 152.·11-s + 325.·13-s − 503.·15-s − 664.·17-s + 1.59e3·19-s + 1.69e3·21-s − 719.·23-s − 2.12e3·25-s + 3.68e3·27-s + 841·29-s − 2.05e3·31-s − 2.43e3·33-s − 3.35e3·35-s + 1.49e4·37-s − 5.20e3·39-s + 1.46e4·41-s − 1.02e4·43-s + 377.·45-s − 4.58e3·47-s − 5.50e3·49-s + 1.06e4·51-s + 8.95e3·53-s + 4.82e3·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 0.564·5-s − 0.819·7-s + 0.0492·9-s + 0.380·11-s + 0.534·13-s − 0.578·15-s − 0.558·17-s + 1.01·19-s + 0.839·21-s − 0.283·23-s − 0.681·25-s + 0.973·27-s + 0.185·29-s − 0.384·31-s − 0.390·33-s − 0.462·35-s + 1.79·37-s − 0.547·39-s + 1.36·41-s − 0.849·43-s + 0.0278·45-s − 0.302·47-s − 0.327·49-s + 0.571·51-s + 0.437·53-s + 0.214·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 841T \) |
| good | 3 | \( 1 + 15.9T + 243T^{2} \) |
| 5 | \( 1 - 31.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 106.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 152.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 325.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 664.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.59e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 719.T + 6.43e6T^{2} \) |
| 31 | \( 1 + 2.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.02e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.58e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.95e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.39e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.98e4T + 8.58e9T^{2} \) |
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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
−9.771374987693152194235513951364, −9.192625847433824475752707995630, −7.897565469062402444864354158883, −6.61187018481029498430917019429, −6.09745261508418940010276343500, −5.27662423201021054303086923108, −3.99497442748019539191717137856, −2.70902907242585623920644654995, −1.18838873375580979315191125233, 0,
1.18838873375580979315191125233, 2.70902907242585623920644654995, 3.99497442748019539191717137856, 5.27662423201021054303086923108, 6.09745261508418940010276343500, 6.61187018481029498430917019429, 7.897565469062402444864354158883, 9.192625847433824475752707995630, 9.771374987693152194235513951364
