| L(s) = 1 | − 2·2-s − 5·5-s + 34·7-s + 8·8-s + 10·10-s + 48·11-s + 70·13-s − 68·14-s − 16·16-s − 54·17-s + 238·19-s − 96·22-s + 51·23-s − 140·26-s + 30·29-s + 133·31-s + 108·34-s − 170·35-s + 436·37-s − 476·38-s − 40·40-s − 156·41-s + 88·43-s − 102·46-s + 516·47-s + 343·49-s − 1.27e3·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 1.83·7-s + 0.353·8-s + 0.316·10-s + 1.31·11-s + 1.49·13-s − 1.29·14-s − 1/4·16-s − 0.770·17-s + 2.87·19-s − 0.930·22-s + 0.462·23-s − 1.05·26-s + 0.192·29-s + 0.770·31-s + 0.544·34-s − 0.821·35-s + 1.93·37-s − 2.03·38-s − 0.158·40-s − 0.594·41-s + 0.312·43-s − 0.326·46-s + 1.60·47-s + 49-s − 3.31·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.028003983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.028003983\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 34 T + 813 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 48 T + 973 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 27 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 119 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 51 T - 9566 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 30 T - 23489 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 133 T - 12102 T^{2} - 133 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 218 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 156 T - 44585 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 88 T - 71763 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 516 T + 162433 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 639 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 654 T + 222337 T^{2} - 654 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 461 T - 14460 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 182 T - 267639 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 900 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 704 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1375 T + 1397586 T^{2} - 1375 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 915 T + 265438 T^{2} + 915 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1116 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T - 912417 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.780435008188312429691570166578, −9.553617964229525615629917322130, −9.288733387513740991426759144884, −8.795758247404053734414515079478, −8.213606931229618715414666269722, −8.047467793223057537072276486770, −7.78002956186958079586354196514, −7.21713342306275261329514937547, −6.51198395646339995088682353326, −6.45811898663149147702034634125, −5.41842260251515926456935485159, −5.33262378440303207895239616333, −4.52277235747127740869478812160, −4.32536399348400064395568786174, −3.63702256694718906086734680340, −3.17878827237601185880042427678, −2.30462108874290371171124459190, −1.43215379251411720581210866643, −1.15382961927888633785313073831, −0.77134806124100156229110670383,
0.77134806124100156229110670383, 1.15382961927888633785313073831, 1.43215379251411720581210866643, 2.30462108874290371171124459190, 3.17878827237601185880042427678, 3.63702256694718906086734680340, 4.32536399348400064395568786174, 4.52277235747127740869478812160, 5.33262378440303207895239616333, 5.41842260251515926456935485159, 6.45811898663149147702034634125, 6.51198395646339995088682353326, 7.21713342306275261329514937547, 7.78002956186958079586354196514, 8.047467793223057537072276486770, 8.213606931229618715414666269722, 8.795758247404053734414515079478, 9.288733387513740991426759144884, 9.553617964229525615629917322130, 9.780435008188312429691570166578
