L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 3·11-s + 12-s − 4·13-s + 14-s + 16-s + 3·17-s − 18-s − 21-s + 3·22-s + 6·23-s − 24-s + 4·26-s + 27-s − 28-s + 4·31-s − 32-s − 3·33-s − 3·34-s + 36-s + 8·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.218·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 0.718·31-s − 0.176·32-s − 0.522·33-s − 0.514·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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
−12.70565822247796, −12.24386482469824, −11.84922134694148, −11.18240011726479, −10.89699632587768, −10.20697145149321, −9.839214985146656, −9.757316387510706, −9.039114417684019, −8.665363143456744, −8.116546892455113, −7.664862156603544, −7.447473639354725, −6.790880949862512, −6.468146546181116, −5.728141438705567, −5.258977534572081, −4.742601369911039, −4.276086342261302, −3.352422059034124, −3.054349362662791, −2.625611722594774, −2.075586453558849, −1.362262434914463, −0.7135698168688426, 0,
0.7135698168688426, 1.362262434914463, 2.075586453558849, 2.625611722594774, 3.054349362662791, 3.352422059034124, 4.276086342261302, 4.742601369911039, 5.258977534572081, 5.728141438705567, 6.468146546181116, 6.790880949862512, 7.447473639354725, 7.664862156603544, 8.116546892455113, 8.665363143456744, 9.039114417684019, 9.757316387510706, 9.839214985146656, 10.20697145149321, 10.89699632587768, 11.18240011726479, 11.84922134694148, 12.24386482469824, 12.70565822247796