| 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 | \( 1 - 6iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.411402410245134482732500493675, −7.71992652527435404944040110652, −6.51131727501405536052947912167, −6.07254718825908925441028359194, −5.50554760272062097990048595291, −4.25646182858396962191431654165, −3.81969958605249550256952489836, −2.49713787351991562700961617916, −1.33684734823920685708670158595, 0,
1.38111541325917976888176293343, 2.91693330284469940768403111698, 3.30307534941128958386480953911, 4.73759518815648592837015290850, 5.20783228605416854432950787906, 6.28622788169727491980378190004, 6.75515450898328673709816814149, 7.41108059342925999216617275842, 8.495378190346036176202606274655
