| L(s) = 1 | + 1.82i·5-s − 7-s + 3.75i·11-s + 3.09i·13-s − 3.61·17-s − 1.65i·19-s − 7.47·23-s + 1.65·25-s + 2.38i·29-s − 8.97·31-s − 1.82i·35-s − 8.94i·37-s − 3.04·41-s − 1.82i·43-s + 6.21·47-s + ⋯ |
| L(s) = 1 | + 0.818i·5-s − 0.377·7-s + 1.13i·11-s + 0.859i·13-s − 0.875·17-s − 0.379i·19-s − 1.55·23-s + 0.330·25-s + 0.443i·29-s − 1.61·31-s − 0.309i·35-s − 1.47i·37-s − 0.475·41-s − 0.278i·43-s + 0.906·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1637269423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1637269423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.82iT - 5T^{2} \) |
| 11 | \( 1 - 3.75iT - 11T^{2} \) |
| 13 | \( 1 - 3.09iT - 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 1.65iT - 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 2.38iT - 29T^{2} \) |
| 31 | \( 1 + 8.97T + 31T^{2} \) |
| 37 | \( 1 + 8.94iT - 37T^{2} \) |
| 41 | \( 1 + 3.04T + 41T^{2} \) |
| 43 | \( 1 + 1.82iT - 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 - 3.21iT - 53T^{2} \) |
| 59 | \( 1 + 12.4iT - 59T^{2} \) |
| 61 | \( 1 - 9.91iT - 61T^{2} \) |
| 67 | \( 1 + 8.73iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 7.24T + 79T^{2} \) |
| 83 | \( 1 + 8.99iT - 83T^{2} \) |
| 89 | \( 1 - 1.54T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.54352668536440832545577281283, −7.18871925898597890141384698321, −6.56282845345648138838219786480, −5.88851805438842068088928240961, −4.89307806059617871229938063361, −4.13466379013916661083364386503, −3.47983738469773811785426830150, −2.31407876473810550175620715479, −1.89246898671864289578532199786, −0.04537146602618752952059079181,
0.962064392076475824609611251743, 2.10153533188236817423854123087, 3.14031449237539099458468686472, 3.85396102786203559086285858067, 4.67517471221604967166401976857, 5.57556667725083032682890619922, 5.97567472392553498709400700039, 6.81077997966603965701405296907, 7.74854845140464639224088249487, 8.378125054376851318824153105038
