| L(s) = 1 | + 38·5-s + 49·7-s + 600·11-s − 674·13-s − 78·17-s + 916·19-s − 4.60e3·23-s − 1.68e3·25-s + 6.81e3·29-s − 7.91e3·31-s + 1.86e3·35-s − 9.27e3·37-s + 242·41-s − 1.11e3·43-s − 2.83e4·47-s + 2.40e3·49-s − 1.02e4·53-s + 2.28e4·55-s − 4.10e3·59-s + 1.58e4·61-s − 2.56e4·65-s + 6.76e4·67-s − 6.74e4·71-s + 1.10e3·73-s + 2.94e4·77-s − 8.41e4·79-s − 2.90e3·83-s + ⋯ |
| L(s) = 1 | + 0.679·5-s + 0.377·7-s + 1.49·11-s − 1.10·13-s − 0.0654·17-s + 0.582·19-s − 1.81·23-s − 0.537·25-s + 1.50·29-s − 1.47·31-s + 0.256·35-s − 1.11·37-s + 0.0224·41-s − 0.0920·43-s − 1.86·47-s + 1/7·49-s − 0.500·53-s + 1.01·55-s − 0.153·59-s + 0.546·61-s − 0.751·65-s + 1.84·67-s − 1.58·71-s + 0.0242·73-s + 0.565·77-s − 1.51·79-s − 0.0463·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 38 T + p^{5} T^{2} \) |
| 11 | \( 1 - 600 T + p^{5} T^{2} \) |
| 13 | \( 1 + 674 T + p^{5} T^{2} \) |
| 17 | \( 1 + 78 T + p^{5} T^{2} \) |
| 19 | \( 1 - 916 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4604 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6810 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7912 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9274 T + p^{5} T^{2} \) |
| 41 | \( 1 - 242 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1116 T + p^{5} T^{2} \) |
| 47 | \( 1 + 28312 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10230 T + p^{5} T^{2} \) |
| 59 | \( 1 + 4108 T + p^{5} T^{2} \) |
| 61 | \( 1 - 15878 T + p^{5} T^{2} \) |
| 67 | \( 1 - 67668 T + p^{5} T^{2} \) |
| 71 | \( 1 + 67492 T + p^{5} T^{2} \) |
| 73 | \( 1 - 1106 T + p^{5} T^{2} \) |
| 79 | \( 1 + 84152 T + p^{5} T^{2} \) |
| 83 | \( 1 + 2908 T + p^{5} T^{2} \) |
| 89 | \( 1 - 8322 T + p^{5} T^{2} \) |
| 97 | \( 1 - 130810 T + p^{5} T^{2} \) |
| 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
−8.912331079933913005308287498760, −8.010144012959440302638610937034, −7.05036580712912724379250861029, −6.27717030040681688113331883124, −5.39168518368467703753732738770, −4.44460171741376731347699933358, −3.46924165518821546257535753431, −2.13326633779411250679509862558, −1.43815171437884425253244085221, 0,
1.43815171437884425253244085221, 2.13326633779411250679509862558, 3.46924165518821546257535753431, 4.44460171741376731347699933358, 5.39168518368467703753732738770, 6.27717030040681688113331883124, 7.05036580712912724379250861029, 8.010144012959440302638610937034, 8.912331079933913005308287498760
