| L(s) = 1 | − 2-s + 4-s + 10·5-s − 9·8-s − 10·10-s + 22·11-s + 22·13-s − 47·16-s + 116·17-s − 102·19-s + 10·20-s − 22·22-s − 260·23-s + 75·25-s − 22·26-s + 196·29-s − 150·31-s + 103·32-s − 116·34-s − 96·37-s + 102·38-s − 90·40-s − 176·41-s − 344·43-s + 22·44-s + 260·46-s + 560·47-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1/8·4-s + 0.894·5-s − 0.397·8-s − 0.316·10-s + 0.603·11-s + 0.469·13-s − 0.734·16-s + 1.65·17-s − 1.23·19-s + 0.111·20-s − 0.213·22-s − 2.35·23-s + 3/5·25-s − 0.165·26-s + 1.25·29-s − 0.869·31-s + 0.568·32-s − 0.585·34-s − 0.426·37-s + 0.435·38-s − 0.355·40-s − 0.670·41-s − 1.21·43-s + 0.0753·44-s + 0.833·46-s + 1.73·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 p T + 2718 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22 T + 4450 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 116 T + 9030 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 102 T + 13134 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 260 T + 34734 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 196 T + 20942 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 150 T + 36542 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 96 T + 82550 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 176 T - 16914 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 p T + 171958 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 560 T + 248606 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 326 T + 204138 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 844 T + 474182 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 204 T + 455006 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 104 T + 537670 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1670 T + 1384382 T^{2} + 1670 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 386 T + 152218 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 928 T + 600710 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 588 T + 1495334 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 522 T + 291282 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−8.417399175437126460832182906878, −8.303051664168433483619884952271, −7.79367836251768885556992224732, −7.41075753999081805837649244010, −6.86169047475777195792345700455, −6.37518980802705233626552324238, −6.23158719810207553787429696299, −5.97264994613631742559995850922, −5.29522120612836792063810024495, −5.13052200273767844145512290324, −4.19960624411151701113767831248, −4.19128463776645862515108061213, −3.57217315147426543463037173250, −2.95031481132222464788993924340, −2.58473619934838414296704277849, −1.91716945934200517257247153403, −1.54223710888771067470345598373, −1.13811492539434948809810250831, 0, 0,
1.13811492539434948809810250831, 1.54223710888771067470345598373, 1.91716945934200517257247153403, 2.58473619934838414296704277849, 2.95031481132222464788993924340, 3.57217315147426543463037173250, 4.19128463776645862515108061213, 4.19960624411151701113767831248, 5.13052200273767844145512290324, 5.29522120612836792063810024495, 5.97264994613631742559995850922, 6.23158719810207553787429696299, 6.37518980802705233626552324238, 6.86169047475777195792345700455, 7.41075753999081805837649244010, 7.79367836251768885556992224732, 8.303051664168433483619884952271, 8.417399175437126460832182906878
