| L(s) = 1 | + 4·2-s + 24·5-s − 77·7-s − 64·8-s + 96·10-s + 408·11-s − 89·13-s − 308·14-s − 256·16-s − 4.17e3·17-s − 5.23e3·19-s + 1.63e3·22-s + 1.75e3·23-s + 3.12e3·25-s − 356·26-s − 7.29e3·29-s − 2.34e3·31-s − 1.67e4·34-s − 1.84e3·35-s − 9.98e3·37-s − 2.09e4·38-s − 1.53e3·40-s − 6.52e3·41-s + 6.23e3·43-s + 7.00e3·46-s − 2.98e4·47-s + 1.68e4·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.429·5-s − 0.593·7-s − 0.353·8-s + 0.303·10-s + 1.01·11-s − 0.146·13-s − 0.419·14-s − 1/4·16-s − 3.50·17-s − 3.32·19-s + 0.718·22-s + 0.690·23-s + 25-s − 0.103·26-s − 1.61·29-s − 0.438·31-s − 2.47·34-s − 0.254·35-s − 1.19·37-s − 2.35·38-s − 0.151·40-s − 0.606·41-s + 0.513·43-s + 0.488·46-s − 1.96·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 24 T - 2549 T^{2} - 24 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 p T - 222 p^{2} T^{2} + 11 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 408 T + 5413 T^{2} - 408 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 89 T - 363372 T^{2} + 89 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2088 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2617 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1752 T - 3366839 T^{2} - 1752 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7296 T + 32720467 T^{2} + 7296 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2348 T - 23116047 T^{2} + 2348 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4993 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6528 T - 73241417 T^{2} + 6528 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6232 T - 108170619 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 29832 T + 660603217 T^{2} + 29832 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 22608 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 19608 T - 330450635 T^{2} - 19608 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 22045 T - 358614276 T^{2} - 22045 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 48131 T + 966468054 T^{2} + 48131 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 720 p T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 30737 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 38219 T - 1616364438 T^{2} + 38219 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 8112 T - 3873236099 T^{2} - 8112 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 44280 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 136651 T + 10086155544 T^{2} - 136651 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.71939401381019862494021256588, −11.14978676099283598083024944575, −10.72787808888959864208771218526, −10.53449189431062653759555245347, −9.322459829523990415534295563701, −9.266913175258583292556952025994, −8.606114153142367913290461690453, −8.477364997328223862576184199591, −7.03602722111364575356220233137, −6.76474793583775348496268593657, −6.40405203410603607550765064233, −5.92389229541271641509264368187, −4.82539971439678838390620426102, −4.47113289333526179591238391904, −3.99607647962446424866527618813, −3.14296442507508825952792166172, −2.06363008731956258365782198237, −1.92326781772676645456015035036, 0, 0,
1.92326781772676645456015035036, 2.06363008731956258365782198237, 3.14296442507508825952792166172, 3.99607647962446424866527618813, 4.47113289333526179591238391904, 4.82539971439678838390620426102, 5.92389229541271641509264368187, 6.40405203410603607550765064233, 6.76474793583775348496268593657, 7.03602722111364575356220233137, 8.477364997328223862576184199591, 8.606114153142367913290461690453, 9.266913175258583292556952025994, 9.322459829523990415534295563701, 10.53449189431062653759555245347, 10.72787808888959864208771218526, 11.14978676099283598083024944575, 11.71939401381019862494021256588
