| L(s) = 1 | + 4·2-s − 8.14·3-s + 16·4-s − 9.33·5-s − 32.5·6-s − 162.·7-s + 64·8-s − 176.·9-s − 37.3·10-s − 297.·11-s − 130.·12-s − 155.·13-s − 649.·14-s + 76.0·15-s + 256·16-s − 706.·18-s − 1.77e3·19-s − 149.·20-s + 1.32e3·21-s − 1.18e3·22-s + 384.·23-s − 521.·24-s − 3.03e3·25-s − 622.·26-s + 3.41e3·27-s − 2.59e3·28-s + 5.37e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.522·3-s + 0.5·4-s − 0.166·5-s − 0.369·6-s − 1.25·7-s + 0.353·8-s − 0.726·9-s − 0.118·10-s − 0.740·11-s − 0.261·12-s − 0.255·13-s − 0.885·14-s + 0.0873·15-s + 0.250·16-s − 0.513·18-s − 1.12·19-s − 0.0834·20-s + 0.654·21-s − 0.523·22-s + 0.151·23-s − 0.184·24-s − 0.972·25-s − 0.180·26-s + 0.902·27-s − 0.626·28-s + 1.18·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.137300670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.137300670\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 8.14T + 243T^{2} \) |
| 5 | \( 1 + 9.33T + 3.12e3T^{2} \) |
| 7 | \( 1 + 162.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 297.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 155.T + 3.71e5T^{2} \) |
| 19 | \( 1 + 1.77e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 384.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.94e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.60e4T + 8.58e9T^{2} \) |
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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
−10.29139465462440834952986583656, −9.087400025915912807397033589679, −8.073364736035876691663650073027, −6.92152763816102184654028460989, −6.17865110739405536199349297329, −5.44485660952408434113591628668, −4.34949320445707782203233969818, −3.21510563768874298599517597679, −2.36222511913317758405212377524, −0.44859849299769046804554362470,
0.44859849299769046804554362470, 2.36222511913317758405212377524, 3.21510563768874298599517597679, 4.34949320445707782203233969818, 5.44485660952408434113591628668, 6.17865110739405536199349297329, 6.92152763816102184654028460989, 8.073364736035876691663650073027, 9.087400025915912807397033589679, 10.29139465462440834952986583656
