| L(s) = 1 | + 2-s + 3·3-s + 4-s − 5-s + 3·6-s − 7-s + 8-s + 6·9-s − 10-s + 2·11-s + 3·12-s + 13-s − 14-s − 3·15-s + 16-s + 3·17-s + 6·18-s − 6·19-s − 20-s − 3·21-s + 2·22-s − 4·23-s + 3·24-s − 4·25-s + 26-s + 9·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s + 0.603·11-s + 0.866·12-s + 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 1.41·18-s − 1.37·19-s − 0.223·20-s − 0.654·21-s + 0.426·22-s − 0.834·23-s + 0.612·24-s − 4/5·25-s + 0.196·26-s + 1.73·27-s − 0.188·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.93120652266592, −14.35856885746393, −14.02577208176228, −13.41967723693243, −13.07938065522112, −12.47625937424472, −12.07285011428199, −11.38911416898799, −10.79015201336829, −10.07156335665007, −9.583871947498465, −9.246075062420785, −8.268604461440865, −8.133339724394346, −7.714855293343772, −6.801169838783432, −6.458685594886003, −5.831142508526029, −4.800125626635407, −4.211625428362740, −3.862647425232963, −3.156690987086880, −2.870590532604897, −1.803519050562718, −1.565864601763068, 0,
1.565864601763068, 1.803519050562718, 2.870590532604897, 3.156690987086880, 3.862647425232963, 4.211625428362740, 4.800125626635407, 5.831142508526029, 6.458685594886003, 6.801169838783432, 7.714855293343772, 8.133339724394346, 8.268604461440865, 9.246075062420785, 9.583871947498465, 10.07156335665007, 10.79015201336829, 11.38911416898799, 12.07285011428199, 12.47625937424472, 13.07938065522112, 13.41967723693243, 14.02577208176228, 14.35856885746393, 14.93120652266592
