| L(s) = 1 | − 5·5-s − 2·7-s + 30·11-s − 4·13-s − 90·17-s + 28·19-s + 120·23-s + 25·25-s − 210·29-s + 4·31-s + 10·35-s + 200·37-s − 240·41-s + 136·43-s − 120·47-s − 339·49-s + 30·53-s − 150·55-s − 450·59-s − 166·61-s + 20·65-s − 908·67-s − 1.02e3·71-s − 250·73-s − 60·77-s + 916·79-s − 1.14e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.107·7-s + 0.822·11-s − 0.0853·13-s − 1.28·17-s + 0.338·19-s + 1.08·23-s + 1/5·25-s − 1.34·29-s + 0.0231·31-s + 0.0482·35-s + 0.888·37-s − 0.914·41-s + 0.482·43-s − 0.372·47-s − 0.988·49-s + 0.0777·53-s − 0.367·55-s − 0.992·59-s − 0.348·61-s + 0.0381·65-s − 1.65·67-s − 1.70·71-s − 0.400·73-s − 0.0888·77-s + 1.30·79-s − 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 90 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 - 4 T + p^{3} T^{2} \) |
| 37 | \( 1 - 200 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 136 T + p^{3} T^{2} \) |
| 47 | \( 1 + 120 T + p^{3} T^{2} \) |
| 53 | \( 1 - 30 T + p^{3} T^{2} \) |
| 59 | \( 1 + 450 T + p^{3} T^{2} \) |
| 61 | \( 1 + 166 T + p^{3} T^{2} \) |
| 67 | \( 1 + 908 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 + 250 T + p^{3} T^{2} \) |
| 79 | \( 1 - 916 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1140 T + p^{3} T^{2} \) |
| 89 | \( 1 - 420 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1538 T + p^{3} T^{2} \) |
| 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
−9.346980995270725877674949224494, −8.926500825570793465231849210962, −7.78133724762294995728148982764, −6.96406946409232727859447992629, −6.13718143066856658872062591097, −4.89932660772051350789148431652, −4.01921360279735807636711056315, −2.93428528686325102374870106414, −1.50346375617115100872645591259, 0,
1.50346375617115100872645591259, 2.93428528686325102374870106414, 4.01921360279735807636711056315, 4.89932660772051350789148431652, 6.13718143066856658872062591097, 6.96406946409232727859447992629, 7.78133724762294995728148982764, 8.926500825570793465231849210962, 9.346980995270725877674949224494
