| L(s) = 1 | − 8·2-s + 48·4-s + 12·5-s − 14·7-s − 256·8-s − 96·10-s − 270·11-s + 688·13-s + 112·14-s + 1.28e3·16-s − 3.13e3·17-s − 134·19-s + 576·20-s + 2.16e3·22-s + 2.21e3·23-s + 2.14e3·25-s − 5.50e3·26-s − 672·28-s − 4.46e3·29-s − 3.36e3·31-s − 6.14e3·32-s + 2.51e4·34-s − 168·35-s − 2.09e4·37-s + 1.07e3·38-s − 3.07e3·40-s − 8.44e3·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.214·5-s − 0.107·7-s − 1.41·8-s − 0.303·10-s − 0.672·11-s + 1.12·13-s + 0.152·14-s + 5/4·16-s − 2.63·17-s − 0.0851·19-s + 0.321·20-s + 0.951·22-s + 0.872·23-s + 0.687·25-s − 1.59·26-s − 0.161·28-s − 0.985·29-s − 0.629·31-s − 1.06·32-s + 3.72·34-s − 0.0231·35-s − 2.51·37-s + 0.120·38-s − 0.303·40-s − 0.784·41-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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 12 T - 2003 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 p T + 25374 T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 270 T + 133102 T^{2} + 270 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 688 T + 653697 T^{2} - 688 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3138 T + 5168851 T^{2} + 3138 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 134 T + 1964358 T^{2} + 134 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2214 T + 6132406 T^{2} - 2214 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4464 T + 43011793 T^{2} + 4464 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 1684 T + p^{5} T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 20936 T + 228365049 T^{2} + 20936 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8448 T + 52972654 T^{2} + 8448 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 26506 T + 378272670 T^{2} - 26506 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 42540 T + 886932190 T^{2} + 42540 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9732 T + 706755598 T^{2} + 9732 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 46764 T + 1354261558 T^{2} + 46764 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16912 T + 14361729 T^{2} - 16912 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 59078 T + 2004316710 T^{2} + 59078 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8346 T + 3479248510 T^{2} - 8346 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 64742 T + 2150005623 T^{2} + 64742 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 33014 T + 338746086 T^{2} + 33014 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11004 T + 2629884934 T^{2} + 11004 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 30138 T + 10629157435 T^{2} + 30138 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 120584 T + 20240815662 T^{2} + 120584 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.09504227897844250059101951899, −11.07179411882629554522965291004, −10.97520659970333866625347213974, −10.13992528655917566877124123055, −9.616432398878805496363967066080, −8.989970031345876128240046338158, −8.606403707737093222117151205110, −8.500135027508478748084974812582, −7.42327196442212885366945068578, −7.16088887554644913800151869491, −6.42323742795251018167255646920, −6.14436146299532832425699206953, −5.17348981703699973465108703125, −4.55273137222915807879699217966, −3.47270017874169246407057448260, −2.85680610064565872207494407996, −1.87224999743476579837425309758, −1.48743855198473418497842843475, 0, 0,
1.48743855198473418497842843475, 1.87224999743476579837425309758, 2.85680610064565872207494407996, 3.47270017874169246407057448260, 4.55273137222915807879699217966, 5.17348981703699973465108703125, 6.14436146299532832425699206953, 6.42323742795251018167255646920, 7.16088887554644913800151869491, 7.42327196442212885366945068578, 8.500135027508478748084974812582, 8.606403707737093222117151205110, 8.989970031345876128240046338158, 9.616432398878805496363967066080, 10.13992528655917566877124123055, 10.97520659970333866625347213974, 11.07179411882629554522965291004, 11.09504227897844250059101951899
