| L(s) = 1 | − 2·2-s + 8·3-s − 5·5-s − 16·6-s + 8·8-s + 27·9-s + 10·10-s − 12·11-s − 116·13-s − 40·15-s − 16·16-s − 66·17-s − 54·18-s + 100·19-s + 24·22-s − 132·23-s + 64·24-s + 232·26-s + 136·27-s − 180·29-s + 80·30-s − 152·31-s − 96·33-s + 132·34-s + 34·37-s − 200·38-s − 928·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s − 0.447·5-s − 1.08·6-s + 0.353·8-s + 9-s + 0.316·10-s − 0.328·11-s − 2.47·13-s − 0.688·15-s − 1/4·16-s − 0.941·17-s − 0.707·18-s + 1.20·19-s + 0.232·22-s − 1.19·23-s + 0.544·24-s + 1.74·26-s + 0.969·27-s − 1.15·29-s + 0.486·30-s − 0.880·31-s − 0.506·33-s + 0.665·34-s + 0.151·37-s − 0.853·38-s − 3.81·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 8 T + 37 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 66 T - 557 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 100 T + 3141 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 132 T + 5257 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 152 T - 6687 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T - 49497 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 438 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 204 T - 62207 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 222 T - 99593 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 420 T - 28979 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 902 T + 586623 T^{2} + 902 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 1024 T + 747813 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 160 T - 467439 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 810 T - 48869 T^{2} + 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1106 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.978219001811952893731562612626, −9.735005175043699495075817388929, −9.530355963939738120997549945111, −8.777830426317841512394121528522, −8.761933759472811732783690841621, −8.001923955291931246624803085316, −7.70374652794096249709160857552, −7.43993759069940494734812899161, −6.98579389977088677360268919126, −6.38249192995478023537377193889, −5.45063535945204850898621580443, −4.87018596648270441465837099525, −4.68048720075219286865180654665, −3.55471375562395561142684249989, −3.51632858072046559963553962464, −2.46258654114247983945173670375, −2.30412967563862227396378281338, −1.49981758617650264657988017142, 0, 0,
1.49981758617650264657988017142, 2.30412967563862227396378281338, 2.46258654114247983945173670375, 3.51632858072046559963553962464, 3.55471375562395561142684249989, 4.68048720075219286865180654665, 4.87018596648270441465837099525, 5.45063535945204850898621580443, 6.38249192995478023537377193889, 6.98579389977088677360268919126, 7.43993759069940494734812899161, 7.70374652794096249709160857552, 8.001923955291931246624803085316, 8.761933759472811732783690841621, 8.777830426317841512394121528522, 9.530355963939738120997549945111, 9.735005175043699495075817388929, 9.978219001811952893731562612626
