| L(s) = 1 | − 1.95·2-s + 3-s + 1.83·4-s + 0.208·5-s − 1.95·6-s − 0.474·7-s + 0.319·8-s + 9-s − 0.407·10-s − 1.72·11-s + 1.83·12-s − 5.59·13-s + 0.928·14-s + 0.208·15-s − 4.29·16-s − 8.14·17-s − 1.95·18-s + 1.99·19-s + 0.382·20-s − 0.474·21-s + 3.38·22-s + 8.32·23-s + 0.319·24-s − 4.95·25-s + 10.9·26-s + 27-s − 0.870·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.577·3-s + 0.918·4-s + 0.0930·5-s − 0.799·6-s − 0.179·7-s + 0.112·8-s + 0.333·9-s − 0.128·10-s − 0.521·11-s + 0.530·12-s − 1.55·13-s + 0.248·14-s + 0.0537·15-s − 1.07·16-s − 1.97·17-s − 0.461·18-s + 0.458·19-s + 0.0854·20-s − 0.103·21-s + 0.722·22-s + 1.73·23-s + 0.0652·24-s − 0.991·25-s + 2.14·26-s + 0.192·27-s − 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7955536078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7955536078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 677 | \( 1 + T \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 5 | \( 1 - 0.208T + 5T^{2} \) |
| 7 | \( 1 + 0.474T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 + 8.14T + 17T^{2} \) |
| 19 | \( 1 - 1.99T + 19T^{2} \) |
| 23 | \( 1 - 8.32T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 8.25T + 53T^{2} \) |
| 59 | \( 1 - 8.99T + 59T^{2} \) |
| 61 | \( 1 - 6.45T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 - 9.51T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 9.81T + 97T^{2} \) |
| 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
−9.333659662429951068015914282221, −8.495484096146136140662853545402, −7.65530651003947956855204144842, −7.22248335826798021256429474375, −6.37337174536268023089622943174, −4.96938294011098487443269441159, −4.31886577931477656236952125142, −2.70374080171444571000182436909, −2.22639229531887286092778934394, −0.68777779500632556998515394337,
0.68777779500632556998515394337, 2.22639229531887286092778934394, 2.70374080171444571000182436909, 4.31886577931477656236952125142, 4.96938294011098487443269441159, 6.37337174536268023089622943174, 7.22248335826798021256429474375, 7.65530651003947956855204144842, 8.495484096146136140662853545402, 9.333659662429951068015914282221
