| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 14-s − 15-s − 16-s − 18-s − 21-s + 23-s + 24-s + 2·27-s − 2·29-s + 30-s + 35-s − 40-s − 2·41-s + 42-s − 2·43-s − 45-s − 46-s − 2·47-s − 48-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 14-s − 15-s − 16-s − 18-s − 21-s + 23-s + 24-s + 2·27-s − 2·29-s + 30-s + 35-s − 40-s − 2·41-s + 42-s − 2·43-s − 45-s − 46-s − 2·47-s − 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2644285454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2644285454\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−14.17587321847294524728955915446, −13.02974905280989370564192259687, −12.96999685799762280518674606496, −12.17860024544497028843624594850, −11.42040436361054528192267196081, −11.16716525752643526132281201912, −10.27748392754102519390547451999, −9.877769206049269076046763875976, −9.614591293890291152644650722770, −8.868601555853835613352467318640, −8.416143973274795439126093561837, −8.180747780401275503283824113235, −7.26104126294525014046992686963, −7.10387591591902298203109404153, −6.41661967871024913557960236053, −5.13376199973804018896584366820, −4.57530209214505313204504732398, −3.43782950479146480647989201129, −3.41763793665203501307430262132, −1.84798733176887999444039248112,
1.84798733176887999444039248112, 3.41763793665203501307430262132, 3.43782950479146480647989201129, 4.57530209214505313204504732398, 5.13376199973804018896584366820, 6.41661967871024913557960236053, 7.10387591591902298203109404153, 7.26104126294525014046992686963, 8.180747780401275503283824113235, 8.416143973274795439126093561837, 8.868601555853835613352467318640, 9.614591293890291152644650722770, 9.877769206049269076046763875976, 10.27748392754102519390547451999, 11.16716525752643526132281201912, 11.42040436361054528192267196081, 12.17860024544497028843624594850, 12.96999685799762280518674606496, 13.02974905280989370564192259687, 14.17587321847294524728955915446
