| L(s) = 1 | + 4·2-s + 16·4-s + 51·5-s − 166·7-s + 64·8-s + 204·10-s + 121·11-s + 692·13-s − 664·14-s + 256·16-s + 738·17-s + 1.42e3·19-s + 816·20-s + 484·22-s + 1.77e3·23-s − 524·25-s + 2.76e3·26-s − 2.65e3·28-s + 2.06e3·29-s + 6.24e3·31-s + 1.02e3·32-s + 2.95e3·34-s − 8.46e3·35-s − 1.47e4·37-s + 5.69e3·38-s + 3.26e3·40-s − 5.30e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.912·5-s − 1.28·7-s + 0.353·8-s + 0.645·10-s + 0.301·11-s + 1.13·13-s − 0.905·14-s + 1/4·16-s + 0.619·17-s + 0.904·19-s + 0.456·20-s + 0.213·22-s + 0.701·23-s − 0.167·25-s + 0.803·26-s − 0.640·28-s + 0.455·29-s + 1.16·31-s + 0.176·32-s + 0.437·34-s − 1.16·35-s − 1.77·37-s + 0.639·38-s + 0.322·40-s − 0.492·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.610844769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610844769\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 51 T + p^{5} T^{2} \) |
| 7 | \( 1 + 166 T + p^{5} T^{2} \) |
| 13 | \( 1 - 692 T + p^{5} T^{2} \) |
| 17 | \( 1 - 738 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1424 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1779 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2064 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6245 T + p^{5} T^{2} \) |
| 37 | \( 1 + 14785 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5304 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17798 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17184 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30726 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34989 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45940 T + p^{5} T^{2} \) |
| 67 | \( 1 - 25343 T + p^{5} T^{2} \) |
| 71 | \( 1 + 13311 T + p^{5} T^{2} \) |
| 73 | \( 1 + 53260 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77234 T + p^{5} T^{2} \) |
| 83 | \( 1 + 55014 T + p^{5} T^{2} \) |
| 89 | \( 1 + 125415 T + p^{5} T^{2} \) |
| 97 | \( 1 + 88807 T + p^{5} T^{2} \) |
| 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
−11.83700609171847667105835091274, −10.53917977694137056521358083941, −9.767278741967544879111150732114, −8.751043933413728847364796339505, −7.12119607819788898213010783360, −6.20098821214609026598843539802, −5.45401310655371747322491048529, −3.80131757839069354872828302352, −2.79326841817557275176218411681, −1.15137657999533007682081241226,
1.15137657999533007682081241226, 2.79326841817557275176218411681, 3.80131757839069354872828302352, 5.45401310655371747322491048529, 6.20098821214609026598843539802, 7.12119607819788898213010783360, 8.751043933413728847364796339505, 9.767278741967544879111150732114, 10.53917977694137056521358083941, 11.83700609171847667105835091274
