| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 2·9-s − 10-s + 2·11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 5·17-s + 2·18-s + 20-s − 21-s − 2·22-s − 2·23-s + 24-s − 4·25-s − 26-s + 5·27-s + 28-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.223·20-s − 0.218·21-s − 0.426·22-s − 0.417·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.962·27-s + 0.188·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43706 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.547578100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547578100\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.69627078882649, −14.15268436890897, −13.89528059289698, −12.93885445235287, −12.55198260481664, −11.84744197546600, −11.51998814575483, −11.09689651256239, −10.45413643815788, −10.01712652273443, −9.335842074863889, −9.042584194723948, −8.251982199316520, −7.871895274005805, −7.247186266306231, −6.545398249630327, −6.013256130084825, −5.524048340541050, −5.158534705494529, −4.018093658569057, −3.627709711486860, −2.637046830066941, −2.050067263193013, −1.213286122447098, −0.5755201438954873,
0.5755201438954873, 1.213286122447098, 2.050067263193013, 2.637046830066941, 3.627709711486860, 4.018093658569057, 5.158534705494529, 5.524048340541050, 6.013256130084825, 6.545398249630327, 7.247186266306231, 7.871895274005805, 8.251982199316520, 9.042584194723948, 9.335842074863889, 10.01712652273443, 10.45413643815788, 11.09689651256239, 11.51998814575483, 11.84744197546600, 12.55198260481664, 12.93885445235287, 13.89528059289698, 14.15268436890897, 14.69627078882649
