| L(s) = 1 | − 3·2-s − 2·5-s + 14·7-s + 3·8-s + 6·10-s + 22·11-s − 134·13-s − 42·14-s − 16-s + 26·17-s + 152·19-s − 66·22-s − 178·23-s − 99·25-s + 402·26-s − 190·29-s + 346·31-s + 168·32-s − 78·34-s − 28·35-s − 206·37-s − 456·38-s − 6·40-s − 244·41-s + 242·43-s + 534·46-s − 920·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.178·5-s + 0.755·7-s + 0.132·8-s + 0.189·10-s + 0.603·11-s − 2.85·13-s − 0.801·14-s − 0.0156·16-s + 0.370·17-s + 1.83·19-s − 0.639·22-s − 1.61·23-s − 0.791·25-s + 3.03·26-s − 1.21·29-s + 2.00·31-s + 0.928·32-s − 0.393·34-s − 0.135·35-s − 0.915·37-s − 1.94·38-s − 0.0237·40-s − 0.929·41-s + 0.858·43-s + 1.71·46-s − 2.85·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 103 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 134 T + 8735 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 26 T + 5518 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 p T + 16497 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 178 T + 31330 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 190 T + 24503 T^{2} + 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 346 T + 89178 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 206 T + 74027 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 244 T + 119426 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 242 T + 61730 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 920 T + 417433 T^{2} + 920 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 550 T + 353806 T^{2} - 550 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 412 T + 430069 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 252 T + 318286 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 588233 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 572 T + 616318 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1026 T + 625471 T^{2} + 1026 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 52 T + 985422 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 138 T + 675882 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1512 T + 1596526 T^{2} + 1512 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1106 T + 2130822 T^{2} + 1106 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.850275722600019857445586262582, −9.602036396684893444613730805093, −9.050762787009883833873861241115, −8.573179794888514900965247735255, −7.977559958490869370132602264311, −7.86075308491119998922911810810, −7.48003209521311523840155229203, −6.88670532623123052753168232225, −6.54277824958488414116181866851, −5.66984460530391837765294128787, −5.16675356874704210567766923999, −5.04500182364427914562724302962, −4.20706622855885860721122428724, −3.89409295726334963251950373406, −2.92306714404741700841562893596, −2.50348578530698587819550462608, −1.72988640033133755240877162168, −1.10613199601176443737102633840, 0, 0,
1.10613199601176443737102633840, 1.72988640033133755240877162168, 2.50348578530698587819550462608, 2.92306714404741700841562893596, 3.89409295726334963251950373406, 4.20706622855885860721122428724, 5.04500182364427914562724302962, 5.16675356874704210567766923999, 5.66984460530391837765294128787, 6.54277824958488414116181866851, 6.88670532623123052753168232225, 7.48003209521311523840155229203, 7.86075308491119998922911810810, 7.977559958490869370132602264311, 8.573179794888514900965247735255, 9.050762787009883833873861241115, 9.602036396684893444613730805093, 9.850275722600019857445586262582
