| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 12-s − 2·13-s − 14-s − 15-s + 16-s − 2·17-s + 18-s − 2·19-s + 20-s + 21-s − 4·23-s − 24-s + 25-s − 2·26-s − 27-s − 28-s + 2·29-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 + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.218·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.74569622622527, −14.94469900185504, −14.58650944827813, −13.93875468447752, −13.44687051884854, −12.90230031314818, −12.35552524745323, −12.11206217382393, −11.24015097774382, −10.81950192641847, −10.37877725696853, −9.523289753512282, −9.327655108225370, −8.385359618465758, −7.581016072053008, −7.218486805304928, −6.302014964866674, −6.086974018937587, −5.524939074990029, −4.600609896934694, −4.367953294342056, −3.498230916399818, −2.603647347354134, −2.126051601878012, −1.106452418716088, 0,
1.106452418716088, 2.126051601878012, 2.603647347354134, 3.498230916399818, 4.367953294342056, 4.600609896934694, 5.524939074990029, 6.086974018937587, 6.302014964866674, 7.218486805304928, 7.581016072053008, 8.385359618465758, 9.327655108225370, 9.523289753512282, 10.37877725696853, 10.81950192641847, 11.24015097774382, 12.11206217382393, 12.35552524745323, 12.90230031314818, 13.44687051884854, 13.93875468447752, 14.58650944827813, 14.94469900185504, 15.74569622622527
