| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·5-s − 2·6-s + 7-s − 3·8-s + 9-s − 2·10-s − 11-s + 2·12-s − 4·13-s + 14-s + 4·15-s − 16-s − 4·17-s + 18-s + 2·20-s − 2·21-s − 22-s − 4·23-s + 6·24-s − 25-s − 4·26-s + 4·27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.447·20-s − 0.436·21-s − 0.213·22-s − 0.834·23-s + 1.22·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129437 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129437 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242338002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242338002\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.66333028652122, −12.78134359902578, −12.38805445310821, −12.09667479352278, −11.69298643308291, −11.37064812215586, −10.65010556235270, −10.20866357969419, −9.844403202810306, −8.911731488890208, −8.643798573708511, −8.102273492300362, −7.399506280981107, −7.084672625927840, −6.149206966120866, −6.058404872313416, −5.342527948466537, −4.769559312275262, −4.471401826523330, −4.151940434369713, −3.302488000715570, −2.647123736491947, −2.094431143386409, −0.6618253536142335, −0.5440966486071980,
0.5440966486071980, 0.6618253536142335, 2.094431143386409, 2.647123736491947, 3.302488000715570, 4.151940434369713, 4.471401826523330, 4.769559312275262, 5.342527948466537, 6.058404872313416, 6.149206966120866, 7.084672625927840, 7.399506280981107, 8.102273492300362, 8.643798573708511, 8.911731488890208, 9.844403202810306, 10.20866357969419, 10.65010556235270, 11.37064812215586, 11.69298643308291, 12.09667479352278, 12.38805445310821, 12.78134359902578, 13.66333028652122
