| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 3·11-s + 13-s + 16-s + 5·17-s − 19-s − 2·20-s − 3·22-s − 25-s + 26-s + 29-s − 4·31-s + 32-s + 5·34-s − 2·37-s − 38-s − 2·40-s + 10·41-s − 10·43-s − 3·44-s − 47-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.904·11-s + 0.277·13-s + 1/4·16-s + 1.21·17-s − 0.229·19-s − 0.447·20-s − 0.639·22-s − 1/5·25-s + 0.196·26-s + 0.185·29-s − 0.718·31-s + 0.176·32-s + 0.857·34-s − 0.328·37-s − 0.162·38-s − 0.316·40-s + 1.56·41-s − 1.52·43-s − 0.452·44-s − 0.145·47-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.37460335252981, −16.14727171830748, −15.57441257956135, −14.96410189432702, −14.55229875165439, −13.89899045090730, −13.19862845351861, −12.79161248846554, −12.13154367076409, −11.67981207977643, −11.07605802130154, −10.46415985877291, −9.925754411152571, −9.079021714004809, −8.154778008098907, −7.903774395090671, −7.217945416733368, −6.551503135310189, −5.608617415956514, −5.299862856236496, −4.365939641538751, −3.772092755089723, −3.141276628565804, −2.343864637495128, −1.242729501895119, 0,
1.242729501895119, 2.343864637495128, 3.141276628565804, 3.772092755089723, 4.365939641538751, 5.299862856236496, 5.608617415956514, 6.551503135310189, 7.217945416733368, 7.903774395090671, 8.154778008098907, 9.079021714004809, 9.925754411152571, 10.46415985877291, 11.07605802130154, 11.67981207977643, 12.13154367076409, 12.79161248846554, 13.19862845351861, 13.89899045090730, 14.55229875165439, 14.96410189432702, 15.57441257956135, 16.14727171830748, 16.37460335252981
