| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 2·11-s − 5·13-s − 2·14-s + 16-s + 17-s + 2·22-s − 23-s − 5·25-s − 5·26-s − 2·28-s + 4·31-s + 32-s + 34-s + 4·37-s − 7·41-s − 43-s + 2·44-s − 46-s − 2·47-s − 3·49-s − 5·50-s − 5·52-s − 53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.603·11-s − 1.38·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.426·22-s − 0.208·23-s − 25-s − 0.980·26-s − 0.377·28-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.657·37-s − 1.09·41-s − 0.152·43-s + 0.301·44-s − 0.147·46-s − 0.291·47-s − 3/7·49-s − 0.707·50-s − 0.693·52-s − 0.137·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.44767937184176, −13.22097993785233, −12.57438619801538, −12.14879496906362, −11.89963061784877, −11.33908263834545, −10.82100008591976, −10.06234531110978, −9.745645360063978, −9.565329867989993, −8.743522227694870, −8.093007881444785, −7.722607145443403, −7.075853556630512, −6.543619251795881, −6.364194079251794, −5.514486347000831, −5.186217833598615, −4.539424908968955, −3.991580052055093, −3.488149424826528, −2.906798438176933, −2.310370347339124, −1.765297234571798, −0.8283230601242600, 0,
0.8283230601242600, 1.765297234571798, 2.310370347339124, 2.906798438176933, 3.488149424826528, 3.991580052055093, 4.539424908968955, 5.186217833598615, 5.514486347000831, 6.364194079251794, 6.543619251795881, 7.075853556630512, 7.722607145443403, 8.093007881444785, 8.743522227694870, 9.565329867989993, 9.745645360063978, 10.06234531110978, 10.82100008591976, 11.33908263834545, 11.89963061784877, 12.14879496906362, 12.57438619801538, 13.22097993785233, 13.44767937184176
