| L(s) = 1 | + 60·4-s − 81·9-s + 944·11-s + 2.57e3·16-s + 4.36e3·19-s − 340·29-s + 1.45e4·31-s − 4.86e3·36-s − 3.23e4·41-s + 5.66e4·44-s + 1.61e4·49-s − 6.92e4·59-s − 7.14e4·61-s + 9.31e4·64-s − 1.38e5·71-s + 2.61e5·76-s − 9.52e4·79-s + 6.56e3·81-s + 1.80e5·89-s − 7.64e4·99-s + 1.57e5·101-s − 2.09e5·109-s − 2.04e4·116-s + 3.46e5·121-s + 8.72e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 1/3·9-s + 2.35·11-s + 2.51·16-s + 2.77·19-s − 0.0750·29-s + 2.71·31-s − 5/8·36-s − 3.00·41-s + 4.41·44-s + 0.963·49-s − 2.58·59-s − 2.45·61-s + 2.84·64-s − 3.25·71-s + 5.19·76-s − 1.71·79-s + 1/9·81-s + 2.40·89-s − 0.784·99-s + 1.53·101-s − 1.69·109-s − 0.140·116-s + 2.14·121-s + 5.09·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.336103042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.336103042\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 16190 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 472 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 271990 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 399870 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2180 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12802990 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 170 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7272 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 138667750 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 16198 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 187597030 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 113919390 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 373538790 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 34600 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 35738 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2666934230 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 69088 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 827773490 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 47640 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2401489270 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 90030 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16052296510 T^{2} + p^{10} T^{4} \) |
| 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
−13.70335732361148260037131013197, −13.68696526933450128664366944938, −12.22382841758390872241947352512, −11.85412685908962981633521662603, −11.81442030868693002500685430312, −11.45653330749066990920462193366, −10.39090441454018521331773884115, −10.14661330772699294589501424270, −9.290826461699677145786083715791, −8.783807795276807021443827350822, −7.72974257672250923390545423188, −7.41956249906421760688317207241, −6.47100997083851269845223483942, −6.43811301554369623397990246275, −5.56773330987969319455383823552, −4.51165556046716474740069610212, −3.18968860438128280190551630255, −3.09315912664642323525199436470, −1.52963420133524362028374424109, −1.21832370413861182765712846000,
1.21832370413861182765712846000, 1.52963420133524362028374424109, 3.09315912664642323525199436470, 3.18968860438128280190551630255, 4.51165556046716474740069610212, 5.56773330987969319455383823552, 6.43811301554369623397990246275, 6.47100997083851269845223483942, 7.41956249906421760688317207241, 7.72974257672250923390545423188, 8.783807795276807021443827350822, 9.290826461699677145786083715791, 10.14661330772699294589501424270, 10.39090441454018521331773884115, 11.45653330749066990920462193366, 11.81442030868693002500685430312, 11.85412685908962981633521662603, 12.22382841758390872241947352512, 13.68696526933450128664366944938, 13.70335732361148260037131013197
