| L(s) = 1 | + 2·3-s − 6·5-s − 20·7-s − 23·9-s + 14·11-s − 54·13-s − 12·15-s − 66·17-s + 162·19-s − 40·21-s − 172·23-s − 89·25-s − 100·27-s + 2·29-s + 128·31-s + 28·33-s + 120·35-s − 158·37-s − 108·39-s + 202·41-s − 298·43-s + 138·45-s + 408·47-s + 57·49-s − 132·51-s + 690·53-s − 84·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.536·5-s − 1.07·7-s − 0.851·9-s + 0.383·11-s − 1.15·13-s − 0.206·15-s − 0.941·17-s + 1.95·19-s − 0.415·21-s − 1.55·23-s − 0.711·25-s − 0.712·27-s + 0.0128·29-s + 0.741·31-s + 0.147·33-s + 0.579·35-s − 0.702·37-s − 0.443·39-s + 0.769·41-s − 1.05·43-s + 0.457·45-s + 1.26·47-s + 0.166·49-s − 0.362·51-s + 1.78·53-s − 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 162 T + p^{3} T^{2} \) |
| 23 | \( 1 + 172 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 298 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 690 T + p^{3} T^{2} \) |
| 59 | \( 1 + 322 T + p^{3} T^{2} \) |
| 61 | \( 1 - 298 T + p^{3} T^{2} \) |
| 67 | \( 1 - 202 T + p^{3} T^{2} \) |
| 71 | \( 1 - 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 + 744 T + p^{3} T^{2} \) |
| 83 | \( 1 + 678 T + p^{3} T^{2} \) |
| 89 | \( 1 + 82 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1122 T + p^{3} T^{2} \) |
| 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
−12.15457242286695730911387395819, −11.62955405816231158834436518069, −10.02935753955642259885311034303, −9.269479990022999905048317332152, −8.033627916401084560010221436778, −6.94962877606344074574683033498, −5.60811684336610888884693566072, −3.90180558884246009604784090386, −2.66397320793618286557279327508, 0,
2.66397320793618286557279327508, 3.90180558884246009604784090386, 5.60811684336610888884693566072, 6.94962877606344074574683033498, 8.033627916401084560010221436778, 9.269479990022999905048317332152, 10.02935753955642259885311034303, 11.62955405816231158834436518069, 12.15457242286695730911387395819
