| L(s) = 1 | + 14.2·3-s − 11.7·5-s − 252.·7-s − 40.1·9-s − 382.·11-s + 258.·13-s − 167.·15-s − 138.·17-s + 361·19-s − 3.59e3·21-s + 837.·23-s − 2.98e3·25-s − 4.03e3·27-s + 2.63e3·29-s − 689.·31-s − 5.44e3·33-s + 2.96e3·35-s − 8.72e3·37-s + 3.68e3·39-s − 3.18e3·41-s − 9.84e3·43-s + 471.·45-s + 2.17e4·47-s + 4.67e4·49-s − 1.97e3·51-s + 3.00e4·53-s + 4.48e3·55-s + ⋯ |
| L(s) = 1 | + 0.913·3-s − 0.210·5-s − 1.94·7-s − 0.165·9-s − 0.952·11-s + 0.424·13-s − 0.191·15-s − 0.116·17-s + 0.229·19-s − 1.77·21-s + 0.329·23-s − 0.955·25-s − 1.06·27-s + 0.581·29-s − 0.128·31-s − 0.870·33-s + 0.408·35-s − 1.04·37-s + 0.388·39-s − 0.295·41-s − 0.811·43-s + 0.0347·45-s + 1.43·47-s + 2.78·49-s − 0.106·51-s + 1.46·53-s + 0.200·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 - 14.2T + 243T^{2} \) |
| 5 | \( 1 + 11.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 252.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 382.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 258.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 138.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 837.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 689.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.17e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.26e3T + 8.58e9T^{2} \) |
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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
−13.24771860914390791545130403388, −12.11911966864014127424801663019, −10.48981904889747504786007646588, −9.483410043425175254590731700487, −8.481048622187917052280340443676, −7.15302093299404013312934218709, −5.79761018933897654858569282007, −3.63959709460520848859669032028, −2.69098048023036938534309571596, 0,
2.69098048023036938534309571596, 3.63959709460520848859669032028, 5.79761018933897654858569282007, 7.15302093299404013312934218709, 8.481048622187917052280340443676, 9.483410043425175254590731700487, 10.48981904889747504786007646588, 12.11911966864014127424801663019, 13.24771860914390791545130403388
