L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s − 2·9-s − 11-s − 12-s + 6·13-s − 14-s + 16-s + 5·17-s + 2·18-s − 7·19-s − 21-s + 22-s − 2·23-s + 24-s − 6·26-s + 5·27-s + 28-s + 8·29-s − 2·31-s − 32-s + 33-s − 5·34-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 − 0.301·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.471·18-s − 1.60·19-s − 0.218·21-s + 0.213·22-s − 0.417·23-s + 0.204·24-s − 1.17·26-s + 0.962·27-s + 0.188·28-s + 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.174·33-s − 0.857·34-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
\(L(1)\) |
\(\approx\) |
\(1.612294367\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.612294367\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 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
−13.81173454158424, −13.17034463617507, −12.65401772621015, −12.09036402745468, −11.66991691743125, −11.26100579138305, −10.65057409762603, −10.43445646827835, −10.00531981703625, −9.089672955632690, −8.642043728685581, −8.408143823311346, −7.861373921363903, −7.294619605361152, −6.500302721070563, −6.106496531875544, −5.828089699582147, −5.157932126311928, −4.431440513232577, −3.861194982240933, −3.146426463852233, −2.538340595479960, −1.823949627104819, −1.014774467694617, −0.5608875868425255,
0.5608875868425255, 1.014774467694617, 1.823949627104819, 2.538340595479960, 3.146426463852233, 3.861194982240933, 4.431440513232577, 5.157932126311928, 5.828089699582147, 6.106496531875544, 6.500302721070563, 7.294619605361152, 7.861373921363903, 8.408143823311346, 8.642043728685581, 9.089672955632690, 10.00531981703625, 10.43445646827835, 10.65057409762603, 11.26100579138305, 11.66991691743125, 12.09036402745468, 12.65401772621015, 13.17034463617507, 13.81173454158424