L(s) = 1 | − 3-s + i·5-s − 7-s − 2·9-s + 3·11-s − i·15-s − 6i·17-s − 6i·19-s + 21-s + 6i·23-s − 25-s + 5·27-s + 6i·29-s + 6i·31-s − 3·33-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447i·5-s − 0.377·7-s − 0.666·9-s + 0.904·11-s − 0.258i·15-s − 1.45i·17-s − 1.37i·19-s + 0.218·21-s + 1.25i·23-s − 0.200·25-s + 0.962·27-s + 1.11i·29-s + 1.07i·31-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (-1 - 6i)T \) |
good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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
−8.495378190346036176202606274655, −7.41108059342925999216617275842, −6.75515450898328673709816814149, −6.28622788169727491980378190004, −5.20783228605416854432950787906, −4.73759518815648592837015290850, −3.30307534941128958386480953911, −2.91693330284469940768403111698, −1.38111541325917976888176293343, 0,
1.33684734823920685708670158595, 2.49713787351991562700961617916, 3.81969958605249550256952489836, 4.25646182858396962191431654165, 5.50554760272062097990048595291, 6.07254718825908925441028359194, 6.51131727501405536052947912167, 7.71992652527435404944040110652, 8.411402410245134482732500493675