L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s + 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s − 20-s + 22-s + 6·23-s + 25-s + 2·26-s + 28-s − 6·29-s + 2·31-s + 32-s + 6·34-s − 35-s + 2·37-s − 4·38-s − 40-s − 12·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.524464523\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.524464523\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−7.71205674481894164530920689090, −7.34223598787784368975538133721, −6.44416402888261732093397555016, −5.78498315765151961033768391335, −5.09077563354603289995658044774, −4.32578190164998574619609517599, −3.63349480816926663886233686575, −2.97274816461659283172155327906, −1.86073175045008721576786160541, −0.900144554998059628090772208480,
0.900144554998059628090772208480, 1.86073175045008721576786160541, 2.97274816461659283172155327906, 3.63349480816926663886233686575, 4.32578190164998574619609517599, 5.09077563354603289995658044774, 5.78498315765151961033768391335, 6.44416402888261732093397555016, 7.34223598787784368975538133721, 7.71205674481894164530920689090