| L(s) = 1 | − 2-s − 3·3-s − 3·5-s + 3·6-s + 7-s + 8-s + 6·9-s + 3·10-s + 6·11-s + 2·13-s − 14-s + 9·15-s − 16-s − 6·17-s − 6·18-s − 14·19-s − 3·21-s − 6·22-s − 6·23-s − 3·24-s + 25-s − 2·26-s − 9·27-s − 15·29-s − 9·30-s − 31-s − 18·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.34·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.948·10-s + 1.80·11-s + 0.554·13-s − 0.267·14-s + 2.32·15-s − 1/4·16-s − 1.45·17-s − 1.41·18-s − 3.21·19-s − 0.654·21-s − 1.27·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 1.73·27-s − 2.78·29-s − 1.64·30-s − 0.179·31-s − 3.13·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−10.62108043940049074657150584349, −10.56033277537924069800158751932, −9.503195583626647707080266526248, −9.450187195264585990824451347706, −8.677292287839795598772334090122, −8.444928812939337793360101341032, −8.008972840050733078740072962436, −7.29038347918450508018725824538, −6.94732879534821655955302750296, −6.44388290227135823512277668245, −6.14122045506470045306757991297, −5.68012931751578587431709225936, −4.69139730742457252405064256904, −4.36063974272498269005425692716, −3.97965966764672119768840062102, −3.73424133301301353413046163971, −1.88447900803097074366696213508, −1.69273878723798108628976202305, 0, 0,
1.69273878723798108628976202305, 1.88447900803097074366696213508, 3.73424133301301353413046163971, 3.97965966764672119768840062102, 4.36063974272498269005425692716, 4.69139730742457252405064256904, 5.68012931751578587431709225936, 6.14122045506470045306757991297, 6.44388290227135823512277668245, 6.94732879534821655955302750296, 7.29038347918450508018725824538, 8.008972840050733078740072962436, 8.444928812939337793360101341032, 8.677292287839795598772334090122, 9.450187195264585990824451347706, 9.503195583626647707080266526248, 10.56033277537924069800158751932, 10.62108043940049074657150584349
