| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s − 4·11-s + 12-s − 13-s + 4·14-s + 16-s + 6·17-s + 18-s − 4·19-s + 4·21-s − 4·22-s + 8·23-s + 24-s − 26-s + 27-s + 4·28-s + 29-s + 8·31-s + 32-s − 4·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.872·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.755·28-s + 0.185·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.189660678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.189660678\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.57326600855063, −13.76776408355041, −13.53827566381161, −12.92443386651432, −12.41158536623502, −11.93359130077254, −11.37035461994069, −10.78003337252583, −10.42803560321623, −9.920328966658889, −9.096736801600945, −8.480419024071078, −8.090566166139862, −7.504577870615798, −7.311104431100057, −6.345018568455824, −5.722184363816121, −5.124168283012073, −4.680657370256367, −4.307751357164505, −3.335233441851698, −2.782765352875505, −2.323284636731266, −1.475666216215628, −0.8302460501315557,
0.8302460501315557, 1.475666216215628, 2.323284636731266, 2.782765352875505, 3.335233441851698, 4.307751357164505, 4.680657370256367, 5.124168283012073, 5.722184363816121, 6.345018568455824, 7.311104431100057, 7.504577870615798, 8.090566166139862, 8.480419024071078, 9.096736801600945, 9.920328966658889, 10.42803560321623, 10.78003337252583, 11.37035461994069, 11.93359130077254, 12.41158536623502, 12.92443386651432, 13.53827566381161, 13.76776408355041, 14.57326600855063
