| L(s) = 1 | − 27·9-s − 118·11-s − 218·19-s + 64·29-s + 20·31-s + 234·41-s + 10·49-s − 784·59-s − 1.42e3·61-s − 1.22e3·71-s − 828·79-s + 162·89-s + 3.18e3·99-s − 468·101-s + 2.46e3·109-s + 7.78e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.61e3·169-s + 5.88e3·171-s + ⋯ |
| L(s) = 1 | − 9-s − 3.23·11-s − 2.63·19-s + 0.409·29-s + 0.115·31-s + 0.891·41-s + 0.0291·49-s − 1.72·59-s − 2.98·61-s − 2.04·71-s − 1.17·79-s + 0.192·89-s + 3.23·99-s − 0.461·101-s + 2.16·109-s + 5.84·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.64·169-s + 2.63·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 59 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3610 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9801 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 109 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 13302 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 62102 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 117 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 8470 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 203022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297430 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 392 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 710 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 537517 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 612 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 476633 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1128933 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 81 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 430658 T^{2} + p^{6} T^{4} \) |
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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
−11.67348721665217986390393719824, −11.15393938352522419346794942557, −10.57688960264807055982568372232, −10.49402007663718025376895868752, −10.11515541157881633164754281975, −8.979985282582842188120706929513, −8.845914893732051975166137237585, −8.104698253017225594076831966140, −7.82334988108034490830774446384, −7.38160533140963653868149082931, −6.31954877183554949997475755517, −6.01862108603600452737511220259, −5.40348380460050031519800036264, −4.74456189187046857897397817155, −4.30213702937028984728472696706, −2.86651536518036680132163365638, −2.83505152557841791525048598942, −1.93969814527912622473200520647, 0, 0,
1.93969814527912622473200520647, 2.83505152557841791525048598942, 2.86651536518036680132163365638, 4.30213702937028984728472696706, 4.74456189187046857897397817155, 5.40348380460050031519800036264, 6.01862108603600452737511220259, 6.31954877183554949997475755517, 7.38160533140963653868149082931, 7.82334988108034490830774446384, 8.104698253017225594076831966140, 8.845914893732051975166137237585, 8.979985282582842188120706929513, 10.11515541157881633164754281975, 10.49402007663718025376895868752, 10.57688960264807055982568372232, 11.15393938352522419346794942557, 11.67348721665217986390393719824
