| L(s) = 1 | − 9·4-s − 9·9-s − 22·11-s + 17·16-s − 144·19-s + 108·29-s − 304·31-s + 81·36-s + 188·41-s + 198·44-s − 338·49-s − 40·59-s + 1.14e3·61-s + 423·64-s − 2.18e3·71-s + 1.29e3·76-s + 32·79-s + 81·81-s + 1.93e3·89-s + 198·99-s − 308·109-s − 972·116-s + 363·121-s + 2.73e3·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 9/8·4-s − 1/3·9-s − 0.603·11-s + 0.265·16-s − 1.73·19-s + 0.691·29-s − 1.76·31-s + 3/8·36-s + 0.716·41-s + 0.678·44-s − 0.985·49-s − 0.0882·59-s + 2.39·61-s + 0.826·64-s − 3.65·71-s + 1.95·76-s + 0.0455·79-s + 1/9·81-s + 2.30·89-s + 0.201·99-s − 0.270·109-s − 0.777·116-s + 3/11·121-s + 1.98·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 338 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9822 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19710 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 71030 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 94 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 119770 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 92046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105910 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 570 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 389926 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1092 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 462190 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1005190 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 966 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1548670 T^{2} + p^{6} T^{4} \) |
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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
−9.440500703607838601833324231994, −9.210786834629953667571615997307, −8.792308472501308066430642780616, −8.277197356375204559752429940271, −8.274609561217945122478950031845, −7.32959587541961698901120712372, −7.30574319605047794120331899797, −6.37971387171448441495689688088, −6.20488315901201843628313121095, −5.54935721311541249866284297275, −5.08469504893387818850389417647, −4.72332645246600471890266891900, −4.12871563082940357236909482986, −3.82297902974150445057968474198, −3.11452814803225443667154612252, −2.43348409304544289609941245268, −1.94202808112835304146473418925, −1.01297395746858549631603253306, 0, 0,
1.01297395746858549631603253306, 1.94202808112835304146473418925, 2.43348409304544289609941245268, 3.11452814803225443667154612252, 3.82297902974150445057968474198, 4.12871563082940357236909482986, 4.72332645246600471890266891900, 5.08469504893387818850389417647, 5.54935721311541249866284297275, 6.20488315901201843628313121095, 6.37971387171448441495689688088, 7.30574319605047794120331899797, 7.32959587541961698901120712372, 8.274609561217945122478950031845, 8.277197356375204559752429940271, 8.792308472501308066430642780616, 9.210786834629953667571615997307, 9.440500703607838601833324231994
