| L(s) = 1 | − 4·5-s − 11-s + 6·13-s + 2·19-s + 4·23-s + 11·25-s − 2·29-s − 2·31-s − 2·37-s − 4·43-s − 6·47-s + 2·53-s + 4·55-s + 14·61-s − 24·65-s − 12·67-s + 8·71-s + 4·73-s − 8·79-s + 2·83-s − 14·89-s − 8·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.301·11-s + 1.66·13-s + 0.458·19-s + 0.834·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s − 0.328·37-s − 0.609·43-s − 0.875·47-s + 0.274·53-s + 0.539·55-s + 1.79·61-s − 2.97·65-s − 1.46·67-s + 0.949·71-s + 0.468·73-s − 0.900·79-s + 0.219·83-s − 1.48·89-s − 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−12.72673422956921, −12.54857411414869, −11.69674504386766, −11.59357037116884, −11.13622063840280, −10.78648691052287, −10.30761345537702, −9.639133499561349, −9.059287384544684, −8.600813911191467, −8.239287400065610, −7.942292635965156, −7.247047456755296, −6.972354642517247, −6.484347238822071, −5.747846114619953, −5.302678569915718, −4.741977878723940, −4.111359740871098, −3.824874248704905, −3.171222972987733, −3.041277849713019, −1.985760465798555, −1.257296758009915, −0.7261740857525457, 0,
0.7261740857525457, 1.257296758009915, 1.985760465798555, 3.041277849713019, 3.171222972987733, 3.824874248704905, 4.111359740871098, 4.741977878723940, 5.302678569915718, 5.747846114619953, 6.484347238822071, 6.972354642517247, 7.247047456755296, 7.942292635965156, 8.239287400065610, 8.600813911191467, 9.059287384544684, 9.639133499561349, 10.30761345537702, 10.78648691052287, 11.13622063840280, 11.59357037116884, 11.69674504386766, 12.54857411414869, 12.72673422956921
