| L(s) = 1 | + 3-s − 2·4-s + 5-s + 3·7-s + 9-s + 4·11-s − 2·12-s + 15-s + 4·16-s + 3·17-s + 8·19-s − 2·20-s + 3·21-s + 25-s + 27-s − 6·28-s + 9·29-s − 5·31-s + 4·33-s + 3·35-s − 2·36-s + 9·37-s + 7·41-s − 4·43-s − 8·44-s + 45-s − 2·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.258·15-s + 16-s + 0.727·17-s + 1.83·19-s − 0.447·20-s + 0.654·21-s + 1/5·25-s + 0.192·27-s − 1.13·28-s + 1.67·29-s − 0.898·31-s + 0.696·33-s + 0.507·35-s − 1/3·36-s + 1.47·37-s + 1.09·41-s − 0.609·43-s − 1.20·44-s + 0.149·45-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.383481243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.383481243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.84090507307028456710160148083, −7.47953200977075673945869485545, −6.39266474337588345916140481734, −5.63250647392578322865911031084, −4.90307351654083838200647110312, −4.38823825915169226440286241801, −3.53725605521994980339182259647, −2.81115829937462615855121342453, −1.44916460017560560616018639190, −1.08808844808888442616464545030,
1.08808844808888442616464545030, 1.44916460017560560616018639190, 2.81115829937462615855121342453, 3.53725605521994980339182259647, 4.38823825915169226440286241801, 4.90307351654083838200647110312, 5.63250647392578322865911031084, 6.39266474337588345916140481734, 7.47953200977075673945869485545, 7.84090507307028456710160148083
