| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 11-s + 13-s + 15-s − 17-s + 19-s + 2·21-s + 25-s + 27-s + 8·29-s − 5·31-s − 33-s + 2·35-s + 3·37-s + 39-s + 4·41-s − 9·43-s + 45-s − 6·47-s − 3·49-s − 51-s − 2·53-s − 55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.898·31-s − 0.174·33-s + 0.338·35-s + 0.493·37-s + 0.160·39-s + 0.624·41-s − 1.37·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.140·51-s − 0.274·53-s − 0.134·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.30257318151040, −12.82972454393211, −12.43647208914365, −11.77946739429212, −11.32534825625619, −10.96446241928188, −10.25364425304047, −10.10536195895646, −9.345155461844938, −9.078206178593725, −8.355292719115859, −8.202302075984754, −7.589246032713349, −7.141020410718270, −6.400291903277706, −6.197024865585054, −5.321762352543085, −4.998099964652202, −4.454919508901641, −3.929967627959775, −3.082031565536250, −2.886073647246261, −2.023896680984662, −1.615719877889707, −1.007718325501353, 0,
1.007718325501353, 1.615719877889707, 2.023896680984662, 2.886073647246261, 3.082031565536250, 3.929967627959775, 4.454919508901641, 4.998099964652202, 5.321762352543085, 6.197024865585054, 6.400291903277706, 7.141020410718270, 7.589246032713349, 8.202302075984754, 8.355292719115859, 9.078206178593725, 9.345155461844938, 10.10536195895646, 10.25364425304047, 10.96446241928188, 11.32534825625619, 11.77946739429212, 12.43647208914365, 12.82972454393211, 13.30257318151040
