| L(s) = 1 | + 2·2-s − 5·4-s − 10·5-s − 14·7-s − 12·8-s − 20·10-s + 16·11-s − 76·13-s − 28·14-s − 11·16-s + 124·17-s − 96·19-s + 50·20-s + 32·22-s + 16·23-s + 75·25-s − 152·26-s + 70·28-s − 188·29-s − 120·31-s − 122·32-s + 248·34-s + 140·35-s − 132·37-s − 192·38-s + 120·40-s − 100·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5/8·4-s − 0.894·5-s − 0.755·7-s − 0.530·8-s − 0.632·10-s + 0.438·11-s − 1.62·13-s − 0.534·14-s − 0.171·16-s + 1.76·17-s − 1.15·19-s + 0.559·20-s + 0.310·22-s + 0.145·23-s + 3/5·25-s − 1.14·26-s + 0.472·28-s − 1.20·29-s − 0.695·31-s − 0.673·32-s + 1.25·34-s + 0.676·35-s − 0.586·37-s − 0.819·38-s + 0.474·40-s − 0.380·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 9 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 16 T - 474 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 76 T + 5806 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 124 T + 13638 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 96 T + 13974 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16 T + 15150 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 188 T + 34286 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 120 T + 60590 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 70814 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T + 89142 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 536 T + 200086 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 928 T + 408830 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 884 T + 460350 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1688 T + 1302310 T^{2} + 1688 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 136 T + 540446 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 508 T + 13078 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 432 T + 602142 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 584 T + 1172390 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1404 T + 1802390 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1188 T + 2161254 T^{2} + 1188 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.94134673100186775562682202165, −10.58184402270117112180228495246, −10.07405441550256819965094553309, −9.513269766894992568374061571147, −9.147187989608741493795246390396, −8.793958841921703801403813658371, −7.940027469793387468935475092495, −7.64082229078550582013608267198, −7.14606179275406043849722986945, −6.64470175715975278717112869722, −5.85706181459430471799933668652, −5.42455824634066542670465415236, −4.74262187066406129836835954238, −4.43570930068180126537677482430, −3.55980383521730591711847740597, −3.48151172056530155917647529141, −2.53407874209039872863235914006, −1.48522751218313573985147771566, 0, 0,
1.48522751218313573985147771566, 2.53407874209039872863235914006, 3.48151172056530155917647529141, 3.55980383521730591711847740597, 4.43570930068180126537677482430, 4.74262187066406129836835954238, 5.42455824634066542670465415236, 5.85706181459430471799933668652, 6.64470175715975278717112869722, 7.14606179275406043849722986945, 7.64082229078550582013608267198, 7.940027469793387468935475092495, 8.793958841921703801403813658371, 9.147187989608741493795246390396, 9.513269766894992568374061571147, 10.07405441550256819965094553309, 10.58184402270117112180228495246, 10.94134673100186775562682202165
