| L(s) = 1 | + 2-s − 11·4-s − 3·5-s + 9·7-s − 15·8-s − 3·10-s + 80·11-s + 9·14-s + 61·16-s − 19·17-s + 84·19-s + 33·20-s + 80·22-s − 196·23-s − 239·25-s − 99·28-s + 44·29-s + 86·31-s + 89·32-s − 19·34-s − 27·35-s − 209·37-s + 84·38-s + 45·40-s − 230·41-s + 287·43-s − 880·44-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.37·4-s − 0.268·5-s + 0.485·7-s − 0.662·8-s − 0.0948·10-s + 2.19·11-s + 0.171·14-s + 0.953·16-s − 0.271·17-s + 1.01·19-s + 0.368·20-s + 0.775·22-s − 1.77·23-s − 1.91·25-s − 0.668·28-s + 0.281·29-s + 0.498·31-s + 0.491·32-s − 0.0958·34-s − 0.130·35-s − 0.928·37-s + 0.358·38-s + 0.177·40-s − 0.876·41-s + 1.01·43-s − 3.01·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 248 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 192 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 19 T + 8688 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 196 T + 33326 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 44 T + 10094 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86 T + 56518 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 209 T + 112120 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 287 T + 92698 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 118 T + 297410 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 368 T + 379266 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 68 T + 373930 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 456 T + 542718 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1958 T + 1961238 T^{2} - 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 928 T + 943870 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.010070294823792919159932014861, −8.598454228163027180398436626523, −8.078916271464225045122566790217, −7.78841939270955392669352595838, −7.44753963865920421021343975895, −6.80309779537962226015820848109, −6.30344870591967083347920950880, −6.06558753781027741166273109450, −5.49901558430668191371463091425, −5.17491368247831587147308559098, −4.47698345894573715125797139923, −4.23475697782100907485617697517, −3.81433886243805089861277327875, −3.72241947582491897860110102451, −2.91147160643551037658633816821, −2.13980044257665512243259396753, −1.36796763921283132514883258952, −1.23026557381232837221757028052, 0, 0,
1.23026557381232837221757028052, 1.36796763921283132514883258952, 2.13980044257665512243259396753, 2.91147160643551037658633816821, 3.72241947582491897860110102451, 3.81433886243805089861277327875, 4.23475697782100907485617697517, 4.47698345894573715125797139923, 5.17491368247831587147308559098, 5.49901558430668191371463091425, 6.06558753781027741166273109450, 6.30344870591967083347920950880, 6.80309779537962226015820848109, 7.44753963865920421021343975895, 7.78841939270955392669352595838, 8.078916271464225045122566790217, 8.598454228163027180398436626523, 9.010070294823792919159932014861
