| L(s) = 1 | + 3·2-s − 19·4-s + 87·5-s − 251·7-s − 45·8-s + 261·10-s − 735·11-s − 722·13-s − 753·14-s − 339·16-s + 603·17-s + 722·19-s − 1.65e3·20-s − 2.20e3·22-s + 6.30e3·23-s + 383·25-s − 2.16e3·26-s + 4.76e3·28-s + 9.44e3·29-s − 7.54e3·31-s − 4.65e3·32-s + 1.80e3·34-s − 2.18e4·35-s − 1.85e4·37-s + 2.16e3·38-s − 3.91e3·40-s − 3.01e4·41-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 0.593·4-s + 1.55·5-s − 1.93·7-s − 0.248·8-s + 0.825·10-s − 1.83·11-s − 1.18·13-s − 1.02·14-s − 0.331·16-s + 0.506·17-s + 0.458·19-s − 0.924·20-s − 0.971·22-s + 2.48·23-s + 0.122·25-s − 0.628·26-s + 1.14·28-s + 2.08·29-s − 1.40·31-s − 0.803·32-s + 0.268·34-s − 3.01·35-s − 2.23·37-s + 0.243·38-s − 0.386·40-s − 2.80·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + 7 p^{2} T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 87 T + 7186 T^{2} - 87 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 251 T + 46266 T^{2} + 251 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 735 T + 357670 T^{2} + 735 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 722 T + 375810 T^{2} + 722 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 603 T - 132788 T^{2} - 603 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6306 T + 19942438 T^{2} - 6306 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9444 T + 63074782 T^{2} - 9444 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7544 T + 71436714 T^{2} + 7544 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 18572 T + 213764010 T^{2} + 18572 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 30198 T + 442316146 T^{2} + 30198 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2629 T + 286598418 T^{2} - 2629 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2067 T + 434600308 T^{2} + 2067 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10008 T + 484301302 T^{2} + 10008 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3180 T + 412367926 T^{2} + 3180 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 50975 T + 1873183380 T^{2} + 50975 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 32096 T + 2321063718 T^{2} + 32096 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7812 T + 3549173326 T^{2} - 7812 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 45149 T + 2495573880 T^{2} + 45149 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1570 p T + 7858725630 T^{2} - 1570 p^{6} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 29256 T + 3100137238 T^{2} + 29256 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10836 T + 8060547670 T^{2} + 10836 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 188900 T + 25196489286 T^{2} + 188900 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.89268235894247763816827349576, −10.85549642818496753815199584081, −10.33519954118688642939031669056, −10.16023665889486077053184630590, −9.655322619838590260827980783205, −9.210601987688285261922870783605, −8.830438870424716336831247842524, −7.977940538117414899635153581550, −7.08850399419436441220439494029, −6.90788208055002138449070069779, −6.19876923203487316266464102660, −5.44972163155723971188339063902, −5.04540701150302946529551579347, −4.88518629947862455855117492616, −3.37979343387962141077094708559, −3.11924035485144146233942214336, −2.47747280952919346423391094764, −1.52985270646137823885620169624, 0, 0,
1.52985270646137823885620169624, 2.47747280952919346423391094764, 3.11924035485144146233942214336, 3.37979343387962141077094708559, 4.88518629947862455855117492616, 5.04540701150302946529551579347, 5.44972163155723971188339063902, 6.19876923203487316266464102660, 6.90788208055002138449070069779, 7.08850399419436441220439494029, 7.977940538117414899635153581550, 8.830438870424716336831247842524, 9.210601987688285261922870783605, 9.655322619838590260827980783205, 10.16023665889486077053184630590, 10.33519954118688642939031669056, 10.85549642818496753815199584081, 11.89268235894247763816827349576
