| L(s) = 1 | − 2-s − 2·3-s + 5·5-s + 2·6-s − 8-s + 3·9-s − 5·10-s − 10·11-s − 13-s − 10·15-s − 16-s + 4·17-s − 3·18-s − 4·19-s + 10·22-s + 2·23-s + 2·24-s + 12·25-s + 26-s − 4·27-s − 2·29-s + 10·30-s − 6·31-s + 6·32-s + 20·33-s − 4·34-s − 18·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 2.23·5-s + 0.816·6-s − 0.353·8-s + 9-s − 1.58·10-s − 3.01·11-s − 0.277·13-s − 2.58·15-s − 1/4·16-s + 0.970·17-s − 0.707·18-s − 0.917·19-s + 2.13·22-s + 0.417·23-s + 0.408·24-s + 12/5·25-s + 0.196·26-s − 0.769·27-s − 0.371·29-s + 1.82·30-s − 1.07·31-s + 1.06·32-s + 3.48·33-s − 0.685·34-s − 2.95·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11431161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11431161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - p T + 13 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 29 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 17 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 106 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 91 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 135 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 147 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 119 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 157 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 179 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 28 T + 377 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
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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
−8.514438048126574516222532215940, −8.136499779817752301934991753475, −7.59807259171976434695777947541, −7.18952364515181235998511849359, −7.09868565878067682651738535718, −6.43132888615884340089352717156, −6.03654855040304321751689339463, −5.64717514797860684514395956554, −5.46466365408898291581584095905, −5.37119402719705524147271311114, −4.71124152447525796083686680208, −4.56568220000562648738218456288, −3.57595384745019470383636489028, −3.06816033915159478728750144025, −2.50717594418137143697826850349, −2.21184603950986399153966220368, −1.78880239625634255939639202840, −1.10569477349345212300087567402, 0, 0,
1.10569477349345212300087567402, 1.78880239625634255939639202840, 2.21184603950986399153966220368, 2.50717594418137143697826850349, 3.06816033915159478728750144025, 3.57595384745019470383636489028, 4.56568220000562648738218456288, 4.71124152447525796083686680208, 5.37119402719705524147271311114, 5.46466365408898291581584095905, 5.64717514797860684514395956554, 6.03654855040304321751689339463, 6.43132888615884340089352717156, 7.09868565878067682651738535718, 7.18952364515181235998511849359, 7.59807259171976434695777947541, 8.136499779817752301934991753475, 8.514438048126574516222532215940
