| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s − 2·9-s + 10-s + 2·11-s − 12-s + 13-s + 3·14-s − 15-s + 16-s + 3·17-s − 2·18-s + 20-s − 3·21-s + 2·22-s − 6·23-s − 24-s − 4·25-s + 26-s + 5·27-s + 3·28-s + 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.223·20-s − 0.654·21-s + 0.426·22-s − 1.25·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.962·27-s + 0.566·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.387036799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.387036799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.54670358937597558505690886218, −9.535113060250141036109461404014, −8.373342967914969728100188752283, −7.70239093051095379852384751914, −6.33895221476746844531501592276, −5.87444795069881235921757318675, −4.94656338872407950490486617667, −4.07070110757789664824492907744, −2.66991230698432885978994611145, −1.35551287918471271219119708450,
1.35551287918471271219119708450, 2.66991230698432885978994611145, 4.07070110757789664824492907744, 4.94656338872407950490486617667, 5.87444795069881235921757318675, 6.33895221476746844531501592276, 7.70239093051095379852384751914, 8.373342967914969728100188752283, 9.535113060250141036109461404014, 10.54670358937597558505690886218
