| L(s) = 1 | − 16·2-s − 11·3-s + 192·4-s − 69·5-s + 176·6-s − 348·7-s − 2.04e3·8-s + 205·9-s + 1.10e3·10-s + 2.72e3·11-s − 2.11e3·12-s − 1.41e4·13-s + 5.56e3·14-s + 759·15-s + 2.04e4·16-s − 3.85e4·17-s − 3.28e3·18-s + 1.37e4·19-s − 1.32e4·20-s + 3.82e3·21-s − 4.35e4·22-s − 7.38e4·23-s + 2.25e4·24-s − 1.12e5·25-s + 2.25e5·26-s − 2.72e4·27-s − 6.68e4·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.235·3-s + 3/2·4-s − 0.246·5-s + 0.332·6-s − 0.383·7-s − 1.41·8-s + 0.0937·9-s + 0.349·10-s + 0.616·11-s − 0.352·12-s − 1.78·13-s + 0.542·14-s + 0.0580·15-s + 5/4·16-s − 1.90·17-s − 0.132·18-s + 0.458·19-s − 0.370·20-s + 0.0901·21-s − 0.872·22-s − 1.26·23-s + 0.332·24-s − 1.43·25-s + 2.51·26-s − 0.266·27-s − 0.575·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 11 T - 28 p T^{2} + 11 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 69 T + 117046 T^{2} + 69 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 348 T + 797665 T^{2} + 348 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2723 T + 40823536 T^{2} - 2723 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14113 T + 172916942 T^{2} + 14113 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 38560 T + 1125592633 T^{2} + 38560 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 73897 T + 7536932062 T^{2} + 73897 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 159813 T + 24183839092 T^{2} - 159813 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 259468 T + 71856067166 T^{2} + 259468 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 528168 T + 207353055814 T^{2} + 528168 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1005650 T + 638624808562 T^{2} - 1005650 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 286217 T + 318805457762 T^{2} - 286217 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1397509 T + 1499105746438 T^{2} + 1397509 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1385969 T + 931113363910 T^{2} + 1385969 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2700953 T + 4572129533584 T^{2} - 2700953 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3975947 T + 8214421559736 T^{2} + 3975947 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 134557 T - 597418012966 T^{2} + 134557 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4202740 T + 19347620336530 T^{2} - 4202740 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 900498 T + 18066288071683 T^{2} - 900498 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6893730 T + 49489747567870 T^{2} + 6893730 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2465330 T + 53280336522142 T^{2} + 2465330 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 17431724 T + 152942834308594 T^{2} - 17431724 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6351934 T + 64489429353042 T^{2} - 6351934 p^{7} T^{3} + p^{14} T^{4} \) |
| 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
−14.36230847015076929129336986001, −14.30478324687344801166094475169, −13.03653878327600035753842071550, −12.51620686896748078946837330061, −11.72232363991383297746253034670, −11.50117680716594998557503551685, −10.61041015698977603458636904459, −10.00156392296742469959240522939, −9.379370441334033253613638594488, −9.024774601216368293015170123697, −7.954296107405867270042512792760, −7.49680709304946892783924495257, −6.70690691654445538368764346621, −6.12576380859891401314780614952, −4.90378022821442643145591828475, −3.82034716797509714618683279097, −2.48349440097751772503419524990, −1.71361744150346441600991888415, 0, 0,
1.71361744150346441600991888415, 2.48349440097751772503419524990, 3.82034716797509714618683279097, 4.90378022821442643145591828475, 6.12576380859891401314780614952, 6.70690691654445538368764346621, 7.49680709304946892783924495257, 7.954296107405867270042512792760, 9.024774601216368293015170123697, 9.379370441334033253613638594488, 10.00156392296742469959240522939, 10.61041015698977603458636904459, 11.50117680716594998557503551685, 11.72232363991383297746253034670, 12.51620686896748078946837330061, 13.03653878327600035753842071550, 14.30478324687344801166094475169, 14.36230847015076929129336986001
