| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 6·11-s + 5·13-s − 14-s + 16-s − 6·17-s + 2·19-s − 3·20-s − 6·22-s − 23-s + 4·25-s − 5·26-s + 28-s + 9·29-s − 4·31-s − 32-s + 6·34-s − 3·35-s − 37-s − 2·38-s + 3·40-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 1.80·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.670·20-s − 1.27·22-s − 0.208·23-s + 4/5·25-s − 0.980·26-s + 0.188·28-s + 1.67·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.507·35-s − 0.164·37-s − 0.324·38-s + 0.474·40-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174821723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174821723\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| 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
−8.761854058353298351577108776635, −8.211974984530970543567838017455, −7.30570070759704049888830995880, −6.67228766799739949189885887843, −6.03145394047309159847095673895, −4.58773847424864181350068589967, −3.98507824920492839917952235112, −3.24255084171393312035434815913, −1.75752945682621392985711791063, −0.77918532315201471319183248707,
0.77918532315201471319183248707, 1.75752945682621392985711791063, 3.24255084171393312035434815913, 3.98507824920492839917952235112, 4.58773847424864181350068589967, 6.03145394047309159847095673895, 6.67228766799739949189885887843, 7.30570070759704049888830995880, 8.211974984530970543567838017455, 8.761854058353298351577108776635
