| L(s) = 1 | + 482·9-s − 1.44e3·11-s + 188·19-s + 8.74e3·29-s − 1.24e4·31-s + 2.40e4·41-s − 2.40e3·49-s − 2.48e3·59-s + 1.51e4·61-s − 7.52e4·71-s − 1.24e4·79-s + 1.73e5·81-s + 9.02e4·89-s − 6.94e5·99-s + 9.42e4·101-s + 4.41e5·109-s + 1.23e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.15e5·169-s + ⋯ |
| L(s) = 1 | + 1.98·9-s − 3.58·11-s + 0.119·19-s + 1.93·29-s − 2.33·31-s + 2.23·41-s − 1/7·49-s − 0.0929·59-s + 0.522·61-s − 1.77·71-s − 0.225·79-s + 2.93·81-s + 1.20·89-s − 7.11·99-s + 0.919·101-s + 3.55·109-s + 7.65·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.11·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.069176985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.069176985\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 482 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 720 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2458 p^{2} T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1267198 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 94 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12863470 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4374 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6244 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 22091110 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12006 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 210111286 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 208808882 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 835362790 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 1242 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7592 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1008408790 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 37632 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3965563342 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6248 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7240316770 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 45126 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 5678123230 T^{2} + p^{10} T^{4} \) |
| 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
−9.848457870421664957455084023860, −9.832570293895776412158818852839, −8.958250196990659294690422451671, −8.649360719332333577174406183918, −7.88655519999573719184279539080, −7.73690482657648142927380389997, −7.32822126539191534170519903721, −7.16343352796214164402195215981, −6.30834623855936092748459620186, −5.85450451788696116084132978317, −5.21057904183596230082472839860, −5.05436040062549233605444697164, −4.42484505528440347936946747270, −4.11241372451629510421263495628, −3.08161184393776156385168972222, −2.95206904774621332742341164589, −2.02340381905979202935592871561, −1.92846793092497751727320626817, −0.75869988972999107093360258247, −0.49616688747952715104488793260,
0.49616688747952715104488793260, 0.75869988972999107093360258247, 1.92846793092497751727320626817, 2.02340381905979202935592871561, 2.95206904774621332742341164589, 3.08161184393776156385168972222, 4.11241372451629510421263495628, 4.42484505528440347936946747270, 5.05436040062549233605444697164, 5.21057904183596230082472839860, 5.85450451788696116084132978317, 6.30834623855936092748459620186, 7.16343352796214164402195215981, 7.32822126539191534170519903721, 7.73690482657648142927380389997, 7.88655519999573719184279539080, 8.649360719332333577174406183918, 8.958250196990659294690422451671, 9.832570293895776412158818852839, 9.848457870421664957455084023860
