| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s − 2·9-s − 10-s + 11-s + 12-s − 3·13-s + 15-s + 16-s + 2·18-s + 6·19-s + 20-s − 22-s + 2·23-s − 24-s + 25-s + 3·26-s − 5·27-s − 7·29-s − 30-s − 8·31-s − 32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.832·13-s + 0.258·15-s + 1/4·16-s + 0.471·18-s + 1.37·19-s + 0.223·20-s − 0.213·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.962·27-s − 1.29·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| 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
−7.79521739753837348117836616842, −7.32778413012845597035752372936, −6.59522303852815103527547868095, −5.54172535463613718508105623810, −5.21957994527827334710636586233, −3.81542681673248893062097467112, −3.09454603715138032091863004547, −2.29381822869837459707103905282, −1.43391883342802291865743404144, 0,
1.43391883342802291865743404144, 2.29381822869837459707103905282, 3.09454603715138032091863004547, 3.81542681673248893062097467112, 5.21957994527827334710636586233, 5.54172535463613718508105623810, 6.59522303852815103527547868095, 7.32778413012845597035752372936, 7.79521739753837348117836616842
