| L(s) = 1 | + 4.64·2-s − 10.4·4-s − 9.65·5-s + 199.·7-s − 197.·8-s − 44.8·10-s − 201.·11-s − 120.·13-s + 924.·14-s − 580.·16-s + 2.03e3·17-s + 724.·19-s + 100.·20-s − 934.·22-s − 529·23-s − 3.03e3·25-s − 557.·26-s − 2.07e3·28-s + 6.36e3·29-s + 8.29e3·31-s + 3.60e3·32-s + 9.46e3·34-s − 1.92e3·35-s + 2.04e3·37-s + 3.36e3·38-s + 1.90e3·40-s + 4.18e3·41-s + ⋯ |
| L(s) = 1 | + 0.820·2-s − 0.326·4-s − 0.172·5-s + 1.53·7-s − 1.08·8-s − 0.141·10-s − 0.501·11-s − 0.197·13-s + 1.26·14-s − 0.567·16-s + 1.71·17-s + 0.460·19-s + 0.0563·20-s − 0.411·22-s − 0.208·23-s − 0.970·25-s − 0.161·26-s − 0.501·28-s + 1.40·29-s + 1.55·31-s + 0.623·32-s + 1.40·34-s − 0.265·35-s + 0.245·37-s + 0.378·38-s + 0.187·40-s + 0.388·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.942104365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.942104365\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 4.64T + 32T^{2} \) |
| 5 | \( 1 + 9.65T + 3.12e3T^{2} \) |
| 7 | \( 1 - 199.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 201.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 120.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 724.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.24e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.39e4T + 8.58e9T^{2} \) |
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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
−11.98635364166442172510337411911, −10.67797403470006149779905803847, −9.599426811630707849217663114273, −8.267830874181775096697552739389, −7.68196105445714466256337167939, −5.90173111198958802498479538825, −5.04718619470641639820394358318, −4.18041000905670944580699163653, −2.76424912997954241219363350118, −0.989402597838226547467795908679,
0.989402597838226547467795908679, 2.76424912997954241219363350118, 4.18041000905670944580699163653, 5.04718619470641639820394358318, 5.90173111198958802498479538825, 7.68196105445714466256337167939, 8.267830874181775096697552739389, 9.599426811630707849217663114273, 10.67797403470006149779905803847, 11.98635364166442172510337411911
