| L(s) = 1 | + 0.0838·2-s − 3-s − 1.99·4-s − 0.0838·6-s − 2.34·7-s − 0.334·8-s + 9-s + 1.99·12-s − 5.61·13-s − 0.196·14-s + 3.95·16-s − 2.13·17-s + 0.0838·18-s + 2.17·19-s + 2.34·21-s − 8.11·23-s + 0.334·24-s − 0.470·26-s − 27-s + 4.66·28-s + 5.11·29-s + 4.93·31-s + 1.00·32-s − 0.179·34-s − 1.99·36-s + 5.25·37-s + 0.182·38-s + ⋯ |
| L(s) = 1 | + 0.0592·2-s − 0.577·3-s − 0.996·4-s − 0.0342·6-s − 0.885·7-s − 0.118·8-s + 0.333·9-s + 0.575·12-s − 1.55·13-s − 0.0524·14-s + 0.989·16-s − 0.518·17-s + 0.0197·18-s + 0.499·19-s + 0.511·21-s − 1.69·23-s + 0.0683·24-s − 0.0923·26-s − 0.192·27-s + 0.882·28-s + 0.949·29-s + 0.886·31-s + 0.176·32-s − 0.0307·34-s − 0.332·36-s + 0.864·37-s + 0.0295·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.0838T + 2T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 + 8.11T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 - 6.99T + 43T^{2} \) |
| 47 | \( 1 - 5.75T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 + 6.51T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 2.38T + 89T^{2} \) |
| 97 | \( 1 + 2.10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.51664252187094062925082573842, −6.51657385146993658268925691730, −6.02455568496274869569715480683, −5.30201070525066448433179758514, −4.49032331397897599137781857826, −4.16246894255713600882928071691, −3.06810447728708450164791838097, −2.30743999560770269017013624829, −0.847097247681144722016004660464, 0,
0.847097247681144722016004660464, 2.30743999560770269017013624829, 3.06810447728708450164791838097, 4.16246894255713600882928071691, 4.49032331397897599137781857826, 5.30201070525066448433179758514, 6.02455568496274869569715480683, 6.51657385146993658268925691730, 7.51664252187094062925082573842
