| L(s) = 1 | − 20·2-s − 20·3-s + 120·4-s + 400·6-s + 100·7-s + 640·8-s + 790·9-s + 4.54e3·11-s − 2.40e3·12-s − 3.54e3·13-s − 2.00e3·14-s − 1.15e4·16-s + 2.73e4·17-s − 1.58e4·18-s + 3.87e4·19-s − 2.00e3·21-s − 9.08e4·22-s + 1.24e5·23-s − 1.28e4·24-s + 7.08e4·26-s − 6.73e4·27-s + 1.20e4·28-s − 7.22e4·29-s + 3.06e5·31-s + 7.29e4·32-s − 9.08e4·33-s − 5.46e5·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.427·3-s + 0.937·4-s + 0.756·6-s + 0.110·7-s + 0.441·8-s + 0.361·9-s + 1.02·11-s − 0.400·12-s − 0.446·13-s − 0.194·14-s − 0.707·16-s + 1.34·17-s − 0.638·18-s + 1.29·19-s − 0.0471·21-s − 1.81·22-s + 2.12·23-s − 0.189·24-s + 0.789·26-s − 0.658·27-s + 0.103·28-s − 0.550·29-s + 1.84·31-s + 0.393·32-s − 0.440·33-s − 2.38·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7526819524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7526819524\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 5 p^{2} T + 35 p^{3} T^{2} + 5 p^{9} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 20 T - 130 p T^{2} + 20 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 100 T + 1411250 T^{2} - 100 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4544 T + 31976326 T^{2} - 4544 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3540 T + 100535470 T^{2} + 3540 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 27340 T + 901005190 T^{2} - 27340 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2040 p T + 113449762 p T^{2} - 2040 p^{8} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 124140 T + 10649684530 T^{2} - 124140 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 72260 T + 6846819118 T^{2} + 72260 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 306824 T + 77964629966 T^{2} - 306824 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 123020 T + 144088599870 T^{2} - 123020 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 264364 T + 161786388886 T^{2} - 264364 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 423300 T + 446651231050 T^{2} + 423300 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 105460 T + 858715356610 T^{2} - 105460 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2391580 T + 3562552504510 T^{2} - 2391580 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1120120 T + 1362334883638 T^{2} + 1120120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2257044 T + 5613447576526 T^{2} - 2257044 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4516460 T + 16742087664890 T^{2} + 4516460 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 621784 T + 17914494152446 T^{2} - 621784 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4569060 T + 23424949855030 T^{2} + 4569060 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4333040 T + 330830231042 p T^{2} - 4333040 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9793020 T + 59971104320890 T^{2} - 9793020 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6025620 T + 89865866149558 T^{2} - 6025620 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4609540 T + 142930351581510 T^{2} + 4609540 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−16.45570389858501316292858202201, −16.28227778878984763170578910710, −14.96493434470865914425534579469, −14.80743303208370292771912600155, −13.71254450604755438732266872508, −13.18593931280487769310645509357, −12.03482885294758751663402442479, −11.77457545858674323883667981207, −10.84506926065475156180025010945, −10.05681379015378481096806964839, −9.556243959847314121512723589077, −9.124198961950055201304904540142, −8.216885360980605847364266559377, −7.52756684251553284047092052593, −6.81653541048283097726178791896, −5.59498244385591433715280723129, −4.61461303766045717788779287552, −3.18976327602921800453378511147, −1.27417314287542001573427560230, −0.76452271497330909346391706013,
0.76452271497330909346391706013, 1.27417314287542001573427560230, 3.18976327602921800453378511147, 4.61461303766045717788779287552, 5.59498244385591433715280723129, 6.81653541048283097726178791896, 7.52756684251553284047092052593, 8.216885360980605847364266559377, 9.124198961950055201304904540142, 9.556243959847314121512723589077, 10.05681379015378481096806964839, 10.84506926065475156180025010945, 11.77457545858674323883667981207, 12.03482885294758751663402442479, 13.18593931280487769310645509357, 13.71254450604755438732266872508, 14.80743303208370292771912600155, 14.96493434470865914425534579469, 16.28227778878984763170578910710, 16.45570389858501316292858202201
