| L(s) = 1 | − 2-s − 4-s + 4·5-s − 4·7-s + 3·8-s − 4·10-s − 4·11-s + 4·14-s − 16-s − 2·17-s − 4·19-s − 4·20-s + 4·22-s − 4·23-s + 11·25-s + 4·28-s + 4·29-s − 2·31-s − 5·32-s + 2·34-s − 16·35-s + 6·37-s + 4·38-s + 12·40-s − 6·41-s − 2·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.51·7-s + 1.06·8-s − 1.26·10-s − 1.20·11-s + 1.06·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.894·20-s + 0.852·22-s − 0.834·23-s + 11/5·25-s + 0.755·28-s + 0.742·29-s − 0.359·31-s − 0.883·32-s + 0.342·34-s − 2.70·35-s + 0.986·37-s + 0.648·38-s + 1.89·40-s − 0.937·41-s − 0.304·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 73 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.76904680886731, −13.53126199896836, −13.27642743469532, −12.75759848188730, −12.50150973186903, −11.57175337288791, −10.68178587026152, −10.42690500461036, −10.02283704343169, −9.773263793844540, −9.155490500977104, −8.815181963921849, −8.259817137779170, −7.636267568003594, −6.879207268518205, −6.471310426517437, −5.960994476662598, −5.504258804580803, −4.846474271584550, −4.328408471959779, −3.434148937890831, −2.804864580621198, −2.209103738138233, −1.712865390126480, −0.6935849740620142, 0,
0.6935849740620142, 1.712865390126480, 2.209103738138233, 2.804864580621198, 3.434148937890831, 4.328408471959779, 4.846474271584550, 5.504258804580803, 5.960994476662598, 6.471310426517437, 6.879207268518205, 7.636267568003594, 8.259817137779170, 8.815181963921849, 9.155490500977104, 9.773263793844540, 10.02283704343169, 10.42690500461036, 10.68178587026152, 11.57175337288791, 12.50150973186903, 12.75759848188730, 13.27642743469532, 13.53126199896836, 13.76904680886731
