| L(s) = 1 | − 4·4-s + 52·13-s + 16·16-s − 42·17-s + 6·23-s − 25·25-s + 300·29-s + 4·43-s + 245·49-s − 208·52-s − 894·53-s − 886·61-s − 64·64-s + 168·68-s − 1.57e3·79-s − 24·92-s + 100·100-s − 1.40e3·101-s − 896·103-s − 342·107-s − 1.40e3·113-s − 1.20e3·116-s + 2.43e3·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.10·13-s + 1/4·16-s − 0.599·17-s + 0.0543·23-s − 1/5·25-s + 1.92·29-s + 0.0141·43-s + 5/7·49-s − 0.554·52-s − 2.31·53-s − 1.85·61-s − 1/8·64-s + 0.299·68-s − 2.23·79-s − 0.0271·92-s + 1/10·100-s − 1.38·101-s − 0.857·103-s − 0.308·107-s − 1.16·113-s − 0.960·116-s + 1.83·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.321173008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321173008\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2437 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 21 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12422 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 15482 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 72065 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 56617 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 124130 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 447 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 369142 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 443 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 269750 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 704797 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 698510 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 785 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1026610 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1332097 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 639425 T^{2} + p^{6} T^{4} \) |
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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
−9.773826659794916421103574821122, −9.028398614923146602334054182863, −8.839379547983269705002282572343, −8.437823014863623164411549969791, −8.069776804037984733425318836123, −7.63910143097913833543684787677, −7.13301876165387065989487904848, −6.56011600019719840309791858287, −6.30899747543768992438457608335, −5.88377409945614653905907302118, −5.36386189368243298349848566330, −4.79018204517322687079805231384, −4.39913213141721199433209562779, −4.08961133608035921227103964918, −3.33862292892492244249358815703, −2.99865332241882154153498704082, −2.39227312855280016271574157308, −1.49311080715617244788414279515, −1.19637272573168887301258227538, −0.29743599960830522458584277805,
0.29743599960830522458584277805, 1.19637272573168887301258227538, 1.49311080715617244788414279515, 2.39227312855280016271574157308, 2.99865332241882154153498704082, 3.33862292892492244249358815703, 4.08961133608035921227103964918, 4.39913213141721199433209562779, 4.79018204517322687079805231384, 5.36386189368243298349848566330, 5.88377409945614653905907302118, 6.30899747543768992438457608335, 6.56011600019719840309791858287, 7.13301876165387065989487904848, 7.63910143097913833543684787677, 8.069776804037984733425318836123, 8.437823014863623164411549969791, 8.839379547983269705002282572343, 9.028398614923146602334054182863, 9.773826659794916421103574821122
