| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 4·7-s + 8-s + 9-s + 2·10-s − 12-s − 2·13-s + 4·14-s − 2·15-s + 16-s − 6·17-s + 18-s + 4·19-s + 2·20-s − 4·21-s + 23-s − 24-s − 25-s − 2·26-s − 27-s + 4·28-s + 29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.872·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 0.185·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.656387065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.656387065\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.263585704963024975830727045832, −7.63156864718409983699746942194, −6.80409326667564881991132180521, −6.04219302925733683856174147153, −5.44674612443065479179177428541, −4.60748063880942386605687322558, −4.36934955938838586718487397549, −2.76749490203986529835135686893, −2.05401603112340154266513492577, −1.09378025244341600177404013908,
1.09378025244341600177404013908, 2.05401603112340154266513492577, 2.76749490203986529835135686893, 4.36934955938838586718487397549, 4.60748063880942386605687322558, 5.44674612443065479179177428541, 6.04219302925733683856174147153, 6.80409326667564881991132180521, 7.63156864718409983699746942194, 8.263585704963024975830727045832
