L(s) = 1 | + (2.10 + 2.10i)5-s + 4.40·7-s + (0.215 − 0.215i)11-s + (2.73 + 2.73i)13-s − 2.36i·17-s + (0.758 − 0.758i)19-s − 1.75i·23-s + 3.86i·25-s + (−5.54 + 5.54i)29-s − 9.01i·31-s + (9.27 + 9.27i)35-s + (−3.10 + 3.10i)37-s − 10.1·41-s + (−3.54 − 3.54i)43-s + 3.90·47-s + ⋯ |
L(s) = 1 | + (0.941 + 0.941i)5-s + 1.66·7-s + (0.0650 − 0.0650i)11-s + (0.758 + 0.758i)13-s − 0.573i·17-s + (0.174 − 0.174i)19-s − 0.366i·23-s + 0.772i·25-s + (−1.02 + 1.02i)29-s − 1.61i·31-s + (1.56 + 1.56i)35-s + (−0.510 + 0.510i)37-s − 1.57·41-s + (−0.540 − 0.540i)43-s + 0.569·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.384466195\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.384466195\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.10 - 2.10i)T + 5iT^{2} \) |
| 7 | \( 1 - 4.40T + 7T^{2} \) |
| 11 | \( 1 + (-0.215 + 0.215i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.73 - 2.73i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.36iT - 17T^{2} \) |
| 19 | \( 1 + (-0.758 + 0.758i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.75iT - 23T^{2} \) |
| 29 | \( 1 + (5.54 - 5.54i)T - 29iT^{2} \) |
| 31 | \( 1 + 9.01iT - 31T^{2} \) |
| 37 | \( 1 + (3.10 - 3.10i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + (3.54 + 3.54i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + (2.71 + 2.71i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.40 - 3.40i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.75 - 1.75i)T + 61iT^{2} \) |
| 67 | \( 1 + (-9.11 + 9.11i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 0.482iT - 73T^{2} \) |
| 79 | \( 1 - 6.88iT - 79T^{2} \) |
| 83 | \( 1 + (4.79 + 4.79i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + 3.34T + 97T^{2} \) |
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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
−9.945846971818488706863983989963, −9.037407561252575420935710469154, −8.287279493787150872343369845994, −7.30552258296550533466281663875, −6.58558550554348564987660284420, −5.61346433631634347394897801106, −4.83477746809332864915723175470, −3.67997628622306965921044598623, −2.33735637647913229537276418839, −1.55021450641754600063936626889,
1.28773583864027717066889447135, 1.92156476934674559115211878850, 3.60130671559037501526696891153, 4.80876765237004896177992913347, 5.34607524848436731397663876070, 6.10103681203030268358041904822, 7.42626164244499811536666182045, 8.325311090271475789145897720399, 8.692175569582537154211754235445, 9.691981499673629779919998796306