L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 5·11-s − 12-s − 4·13-s − 14-s + 16-s − 4·17-s − 18-s − 21-s + 5·22-s − 23-s + 24-s + 4·26-s − 27-s + 28-s − 3·29-s − 6·31-s − 32-s + 5·33-s + 4·34-s + 36-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 + 1/3·9-s − 1.50·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.218·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 0.557·29-s − 1.07·31-s − 0.176·32-s + 0.870·33-s + 0.685·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−12.69291494652945, −12.53231298946248, −11.96809079472159, −11.39190326331304, −11.05339764678673, −10.74371164044563, −10.21378937516689, −9.823339188277833, −9.441206127075664, −8.780898548012714, −8.396681229259091, −7.878562119094827, −7.428944189421076, −7.063897779841582, −6.679812629312195, −5.865532530857356, −5.536648256707415, −5.064578283156910, −4.649606836739604, −4.011449386967711, −3.351981849190877, −2.613324807317149, −2.210354506174237, −1.797741425063670, −0.9139827024298164, 0, 0,
0.9139827024298164, 1.797741425063670, 2.210354506174237, 2.613324807317149, 3.351981849190877, 4.011449386967711, 4.649606836739604, 5.064578283156910, 5.536648256707415, 5.865532530857356, 6.679812629312195, 7.063897779841582, 7.428944189421076, 7.878562119094827, 8.396681229259091, 8.780898548012714, 9.441206127075664, 9.823339188277833, 10.21378937516689, 10.74371164044563, 11.05339764678673, 11.39190326331304, 11.96809079472159, 12.53231298946248, 12.69291494652945