| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s + 7·13-s + 14-s − 15-s + 16-s + 3·17-s + 18-s − 19-s − 20-s + 21-s + 24-s − 4·25-s + 7·26-s + 27-s + 28-s − 9·29-s − 30-s − 8·31-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 + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.94·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.204·24-s − 4/5·25-s + 1.37·26-s + 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−14.20206559882954, −13.47636018911588, −13.03363191563325, −12.88191382398126, −12.05036939677170, −11.60638334927128, −11.11391598345370, −10.82903355927828, −10.16546784311901, −9.564869553538872, −8.831498768521996, −8.652926436894068, −7.884566436614153, −7.539901073343734, −7.104533653557429, −6.203009348587994, −5.910535645422511, −5.350670318631649, −4.642801773840117, −3.968302936483791, −3.516832652281282, −3.384097064785173, −2.302038771225521, −1.690809127763667, −1.211437166964166, 0,
1.211437166964166, 1.690809127763667, 2.302038771225521, 3.384097064785173, 3.516832652281282, 3.968302936483791, 4.642801773840117, 5.350670318631649, 5.910535645422511, 6.203009348587994, 7.104533653557429, 7.539901073343734, 7.884566436614153, 8.652926436894068, 8.831498768521996, 9.564869553538872, 10.16546784311901, 10.82903355927828, 11.11391598345370, 11.60638334927128, 12.05036939677170, 12.88191382398126, 13.03363191563325, 13.47636018911588, 14.20206559882954
