L(s) = 1 | − 0.363i·5-s + 4.38i·7-s − 5.87·11-s + 5.79·13-s + 6.55i·17-s − i·19-s + 1.26·23-s + 4.86·25-s − 5.50i·29-s − 1.79i·31-s + 1.59·35-s − 6.76·37-s + 7.61i·41-s − 3.92i·43-s − 9.53·47-s + ⋯ |
L(s) = 1 | − 0.162i·5-s + 1.65i·7-s − 1.77·11-s + 1.60·13-s + 1.59i·17-s − 0.229i·19-s + 0.263·23-s + 0.973·25-s − 1.02i·29-s − 0.321i·31-s + 0.268·35-s − 1.11·37-s + 1.18i·41-s − 0.598i·43-s − 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.025774720\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.025774720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 + 0.363iT - 5T^{2} \) |
| 7 | \( 1 - 4.38iT - 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 - 5.79T + 13T^{2} \) |
| 17 | \( 1 - 6.55iT - 17T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 5.50iT - 29T^{2} \) |
| 31 | \( 1 + 1.79iT - 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 - 7.61iT - 41T^{2} \) |
| 43 | \( 1 + 3.92iT - 43T^{2} \) |
| 47 | \( 1 + 9.53T + 47T^{2} \) |
| 53 | \( 1 - 4.13iT - 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.895T + 61T^{2} \) |
| 67 | \( 1 - 6.97iT - 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 8.38T + 73T^{2} \) |
| 79 | \( 1 - 16.6iT - 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 8.04iT - 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−8.905378371031988901020276797235, −8.314581786004466933174126043691, −8.038997011454580035973007978292, −6.65515836613838643863961188559, −5.88534605086674002978754163907, −5.47120450968320924776723072151, −4.52884578058922630247946818412, −3.32362933193031467411625013431, −2.56380837113565193380784083861, −1.55145759084189284227040603549,
0.33135807270013148906587928588, 1.46935983266311121854571825014, 3.00800664280868536199790588309, 3.50786772202437299070286264597, 4.71783278374422149571603230226, 5.20536783042937366430790838886, 6.38507401162107184256172532870, 7.12981255507359966578845488848, 7.63784822997327539038198418456, 8.429355017298820266526992666148