L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 8·8-s + 9·9-s − 10·10-s − 19·11-s + 12·12-s + 33·13-s − 15·15-s + 16·16-s + 64·17-s + 18·18-s − 141·19-s − 20·20-s − 38·22-s − 51·23-s + 24·24-s + 25·25-s + 66·26-s + 27·27-s + 216·29-s − 30·30-s + 290·31-s + 32·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.520·11-s + 0.288·12-s + 0.704·13-s − 0.258·15-s + 1/4·16-s + 0.913·17-s + 0.235·18-s − 1.70·19-s − 0.223·20-s − 0.368·22-s − 0.462·23-s + 0.204·24-s + 1/5·25-s + 0.497·26-s + 0.192·27-s + 1.38·29-s − 0.182·30-s + 1.68·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.150580337\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.150580337\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 19 T + p^{3} T^{2} \) |
| 13 | \( 1 - 33 T + p^{3} T^{2} \) |
| 17 | \( 1 - 64 T + p^{3} T^{2} \) |
| 19 | \( 1 + 141 T + p^{3} T^{2} \) |
| 23 | \( 1 + 51 T + p^{3} T^{2} \) |
| 29 | \( 1 - 216 T + p^{3} T^{2} \) |
| 31 | \( 1 - 290 T + p^{3} T^{2} \) |
| 37 | \( 1 + 109 T + p^{3} T^{2} \) |
| 41 | \( 1 - 457 T + p^{3} T^{2} \) |
| 43 | \( 1 - 184 T + p^{3} T^{2} \) |
| 47 | \( 1 - 313 T + p^{3} T^{2} \) |
| 53 | \( 1 + 319 T + p^{3} T^{2} \) |
| 59 | \( 1 - 44 T + p^{3} T^{2} \) |
| 61 | \( 1 + 368 T + p^{3} T^{2} \) |
| 67 | \( 1 - 216 T + p^{3} T^{2} \) |
| 71 | \( 1 + 314 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 - 112 T + p^{3} T^{2} \) |
| 83 | \( 1 - 712 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1018 T + p^{3} T^{2} \) |
| 97 | \( 1 - 584 T + p^{3} T^{2} \) |
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
−8.950630607541310365953373570104, −8.190932487821999292969728141857, −7.66926785566122115648490713777, −6.55288556311377615273636355323, −5.92968895501135191652454313425, −4.69278796364179269200543284498, −4.08319826298312552239206874508, −3.09962742739372842869233495946, −2.26774361137910217209147479399, −0.887651083856097770918255136273,
0.887651083856097770918255136273, 2.26774361137910217209147479399, 3.09962742739372842869233495946, 4.08319826298312552239206874508, 4.69278796364179269200543284498, 5.92968895501135191652454313425, 6.55288556311377615273636355323, 7.66926785566122115648490713777, 8.190932487821999292969728141857, 8.950630607541310365953373570104