| L(s) = 1 | − 2·3-s + 4-s + 4·9-s − 2·12-s − 6·13-s − 3·16-s + 8·23-s + 2·25-s − 5·27-s − 13·29-s + 4·36-s + 12·39-s − 2·43-s + 6·48-s + 49-s − 6·52-s − 5·53-s − 12·61-s − 7·64-s − 16·69-s − 4·75-s + 13·79-s + 4·81-s + 26·87-s + 8·92-s + 2·100-s − 16·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 4/3·9-s − 0.577·12-s − 1.66·13-s − 3/4·16-s + 1.66·23-s + 2/5·25-s − 0.962·27-s − 2.41·29-s + 2/3·36-s + 1.92·39-s − 0.304·43-s + 0.866·48-s + 1/7·49-s − 0.832·52-s − 0.686·53-s − 1.53·61-s − 7/8·64-s − 1.92·69-s − 0.461·75-s + 1.46·79-s + 4/9·81-s + 2.78·87-s + 0.834·92-s + 1/5·100-s − 1.59·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130299 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130299 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 257 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
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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
−9.389434006236662257866462170991, −8.856727366412736884879351798508, −7.897989633982946005800867276360, −7.42601733891164258471787769867, −7.19538406758942105902594100962, −6.63446474334995854148568589639, −6.27779463749130551938602487295, −5.37239450597629614951800168608, −5.17242324946863764798903618907, −4.62441841076814613386562922440, −3.98172620748362877077238450205, −3.05783406874093789686590222193, −2.31682846498223717527349919200, −1.46031728981134736337328990499, 0,
1.46031728981134736337328990499, 2.31682846498223717527349919200, 3.05783406874093789686590222193, 3.98172620748362877077238450205, 4.62441841076814613386562922440, 5.17242324946863764798903618907, 5.37239450597629614951800168608, 6.27779463749130551938602487295, 6.63446474334995854148568589639, 7.19538406758942105902594100962, 7.42601733891164258471787769867, 7.897989633982946005800867276360, 8.856727366412736884879351798508, 9.389434006236662257866462170991
