| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 2·9-s + 12-s − 2·13-s − 14-s + 16-s − 3·17-s − 2·18-s − 21-s + 4·23-s + 24-s − 2·26-s − 5·27-s − 28-s + 6·29-s − 31-s + 32-s − 3·34-s − 2·36-s + 2·37-s − 2·39-s − 5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.218·21-s + 0.834·23-s + 0.204·24-s − 0.392·26-s − 0.962·27-s − 0.188·28-s + 1.11·29-s − 0.179·31-s + 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.328·37-s − 0.320·39-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.912989617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.912989617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
| 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
−13.83239189829702, −13.16842252203741, −12.98529029276860, −12.35855308498339, −11.69546120960056, −11.50395976313002, −10.90200410943016, −10.18470268512724, −9.941209580687936, −9.144402409878931, −8.767514827559357, −8.305063330602999, −7.662926882302832, −7.147504206313644, −6.567386233952250, −6.194617573870398, −5.492239243468348, −4.904274560482063, −4.540732283869592, −3.729852660509003, −3.186635341023450, −2.786166180877153, −2.198303004041140, −1.488149563719011, −0.4288418950615721,
0.4288418950615721, 1.488149563719011, 2.198303004041140, 2.786166180877153, 3.186635341023450, 3.729852660509003, 4.540732283869592, 4.904274560482063, 5.492239243468348, 6.194617573870398, 6.567386233952250, 7.147504206313644, 7.662926882302832, 8.305063330602999, 8.767514827559357, 9.144402409878931, 9.941209580687936, 10.18470268512724, 10.90200410943016, 11.50395976313002, 11.69546120960056, 12.35855308498339, 12.98529029276860, 13.16842252203741, 13.83239189829702
