| L(s) = 1 | − 16·5-s + 12·7-s − 64·11-s − 58·13-s + 64·17-s − 272·19-s + 128·23-s + 125·25-s + 144·29-s − 20·31-s − 192·35-s − 36·37-s + 288·41-s + 200·43-s − 384·47-s + 343·49-s + 992·53-s + 1.02e3·55-s + 128·59-s + 458·61-s + 928·65-s + 496·67-s + 1.02e3·71-s − 1.20e3·73-s − 768·77-s − 1.10e3·79-s − 704·83-s + ⋯ |
| L(s) = 1 | − 1.43·5-s + 0.647·7-s − 1.75·11-s − 1.23·13-s + 0.913·17-s − 3.28·19-s + 1.16·23-s + 25-s + 0.922·29-s − 0.115·31-s − 0.927·35-s − 0.159·37-s + 1.09·41-s + 0.709·43-s − 1.19·47-s + 49-s + 2.57·53-s + 2.51·55-s + 0.282·59-s + 0.961·61-s + 1.77·65-s + 0.904·67-s + 1.71·71-s − 1.93·73-s − 1.13·77-s − 1.57·79-s − 0.931·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3579126440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3579126440\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 16 T + 131 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T - 199 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 64 T + 2765 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 58 T + 1167 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 136 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 128 T + 4217 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 144 T - 3653 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T - 29391 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 288 T + 14023 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 200 T - 39507 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 384 T + 43633 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 496 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 128 T - 188995 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 458 T - 17217 T^{2} - 458 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 496 T - 54747 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 512 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 602 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1108 T + 734625 T^{2} + 1108 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 704 T - 76171 T^{2} + 704 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 960 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 206 T - 870237 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.30317300752612981603751666371, −10.25512477292168568504778283837, −9.584023040577093149018666384182, −8.804070318353789023205797094267, −8.362702465725537388302163764349, −8.341819193560368765286063947817, −7.83603816522834038350724006730, −7.22092105204450207650920847172, −7.11121683836829037290699900243, −6.51589047155888908406421378589, −5.53494014480294392281538811907, −5.47239983871650745135704576087, −4.72189585694596276762886029599, −4.25394546862651660033030232145, −4.10119908136202563379676472327, −3.11219841865055026499303501665, −2.47280668123143754883312515921, −2.29206550522509430282683462076, −1.02902535160579133908424864288, −0.19631785940515196948637598578,
0.19631785940515196948637598578, 1.02902535160579133908424864288, 2.29206550522509430282683462076, 2.47280668123143754883312515921, 3.11219841865055026499303501665, 4.10119908136202563379676472327, 4.25394546862651660033030232145, 4.72189585694596276762886029599, 5.47239983871650745135704576087, 5.53494014480294392281538811907, 6.51589047155888908406421378589, 7.11121683836829037290699900243, 7.22092105204450207650920847172, 7.83603816522834038350724006730, 8.341819193560368765286063947817, 8.362702465725537388302163764349, 8.804070318353789023205797094267, 9.584023040577093149018666384182, 10.25512477292168568504778283837, 10.30317300752612981603751666371
