| L(s) = 1 | − 9·5-s − 19·7-s + 24·11-s + 61·13-s + 3·17-s − 133·19-s − 69·23-s + 47·25-s + 237·29-s − 211·31-s + 171·35-s + 262·37-s + 468·41-s + 86·43-s − 483·47-s − 179·49-s + 150·53-s − 216·55-s − 168·59-s − 1.04e3·61-s − 549·65-s + 1.16e3·67-s + 312·71-s − 311·73-s − 456·77-s − 349·79-s − 1.22e3·83-s + ⋯ |
| L(s) = 1 | − 0.804·5-s − 1.02·7-s + 0.657·11-s + 1.30·13-s + 0.0428·17-s − 1.60·19-s − 0.625·23-s + 0.375·25-s + 1.51·29-s − 1.22·31-s + 0.825·35-s + 1.16·37-s + 1.78·41-s + 0.304·43-s − 1.49·47-s − 0.521·49-s + 0.388·53-s − 0.529·55-s − 0.370·59-s − 2.20·61-s − 1.04·65-s + 2.12·67-s + 0.521·71-s − 0.498·73-s − 0.674·77-s − 0.497·79-s − 1.61·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 9 T + 34 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 19 T + 540 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 24 T + 1861 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 61 T + 5088 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 9592 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 p T + 12234 T^{2} + 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 p T + 25288 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 237 T + 51244 T^{2} - 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 211 T + 68586 T^{2} + 211 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 468 T + 191653 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 p T + 24783 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 483 T + 212812 T^{2} + 483 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 150 T + 257074 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 168 T + 416869 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1049 T + 675906 T^{2} + 1049 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1166 T + 907395 T^{2} - 1166 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 312 T + 498238 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 349 T + 976602 T^{2} + 349 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1221 T + 1412098 T^{2} + 1221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 492 T + 1092454 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 128 T + 1715097 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.041232737197850182583404112239, −8.649943544204584098382883444364, −8.284038546259813695749876712542, −7.925995574502723661052342186677, −7.57353856688698253154020831817, −6.80682818069566440938793149961, −6.48582211783719909102685364995, −6.43945667599684781442461463774, −5.82949742282690336715084147028, −5.42857657445758727538791160747, −4.49460657049168364759214948300, −4.35506580194251090520503546963, −3.77743990886932100527122152903, −3.59923453134988988297591347680, −2.76308468139932512643249557655, −2.52600093387767785139710340973, −1.45398939100813340664998121946, −1.14123685937038946307815989008, 0, 0,
1.14123685937038946307815989008, 1.45398939100813340664998121946, 2.52600093387767785139710340973, 2.76308468139932512643249557655, 3.59923453134988988297591347680, 3.77743990886932100527122152903, 4.35506580194251090520503546963, 4.49460657049168364759214948300, 5.42857657445758727538791160747, 5.82949742282690336715084147028, 6.43945667599684781442461463774, 6.48582211783719909102685364995, 6.80682818069566440938793149961, 7.57353856688698253154020831817, 7.925995574502723661052342186677, 8.284038546259813695749876712542, 8.649943544204584098382883444364, 9.041232737197850182583404112239
