| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 2·11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 18-s + 4·19-s − 20-s + 21-s − 2·22-s − 24-s + 25-s + 26-s + 27-s + 28-s + 2·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52720953039143, −12.24327055557444, −11.80444026585734, −11.24785745864188, −10.91862243757170, −10.51591284384617, −9.727725778146710, −9.619579240307654, −8.914179762581109, −8.834212159669038, −8.135493225369196, −7.647812347748181, −7.387587259315882, −7.022672236336703, −6.275681975445139, −5.901066155279046, −5.194557473508689, −4.717573489341377, −4.136372969760321, −3.551834568980734, −3.188596449552548, −2.520347415113847, −1.934973704464504, −1.389567012460474, −0.8172176009174455, 0,
0.8172176009174455, 1.389567012460474, 1.934973704464504, 2.520347415113847, 3.188596449552548, 3.551834568980734, 4.136372969760321, 4.717573489341377, 5.194557473508689, 5.901066155279046, 6.275681975445139, 7.022672236336703, 7.387587259315882, 7.647812347748181, 8.135493225369196, 8.834212159669038, 8.914179762581109, 9.619579240307654, 9.727725778146710, 10.51591284384617, 10.91862243757170, 11.24785745864188, 11.80444026585734, 12.24327055557444, 12.52720953039143
