| L(s) = 1 | − 2-s − 4·5-s + 7-s + 8-s + 4·10-s − 2·11-s + 12·13-s − 14-s − 16-s − 2·17-s + 4·19-s + 2·22-s + 23-s + 5·25-s − 12·26-s + 8·29-s + 9·31-s + 2·34-s − 4·35-s − 8·37-s − 4·38-s − 4·40-s + 6·41-s + 4·43-s − 46-s + 9·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s − 0.603·11-s + 3.32·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.426·22-s + 0.208·23-s + 25-s − 2.35·26-s + 1.48·29-s + 1.61·31-s + 0.342·34-s − 0.676·35-s − 1.31·37-s − 0.648·38-s − 0.632·40-s + 0.937·41-s + 0.609·43-s − 0.147·46-s + 1.31·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343639026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343639026\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.25737184758959382265477222857, −9.420141844727443610376752719788, −8.916149983533698404039515493193, −8.820794616823318769592956929817, −8.202532682907544812943221800991, −8.181077312189295711908508927066, −7.77924231238583808540504852395, −7.30154233033532838532509207918, −6.82105643504306638255227336053, −6.15213765450410592060512003283, −6.09185949045987581664774918540, −5.27086118095282339132565153868, −4.71096349616497830434552604123, −4.32458217783164232559932871444, −3.69568954276267644779935056547, −3.56859320293782943933975143561, −2.92101296086047482613191160163, −2.02650303208289799089492690495, −0.902563278206519705593890782152, −0.876661825813929335953256138876,
0.876661825813929335953256138876, 0.902563278206519705593890782152, 2.02650303208289799089492690495, 2.92101296086047482613191160163, 3.56859320293782943933975143561, 3.69568954276267644779935056547, 4.32458217783164232559932871444, 4.71096349616497830434552604123, 5.27086118095282339132565153868, 6.09185949045987581664774918540, 6.15213765450410592060512003283, 6.82105643504306638255227336053, 7.30154233033532838532509207918, 7.77924231238583808540504852395, 8.181077312189295711908508927066, 8.202532682907544812943221800991, 8.820794616823318769592956929817, 8.916149983533698404039515493193, 9.420141844727443610376752719788, 10.25737184758959382265477222857
