| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 7·13-s − 14-s + 15-s + 16-s − 4·17-s + 18-s + 5·19-s + 20-s − 21-s + 22-s + 8·23-s + 24-s − 4·25-s − 7·26-s + 27-s − 28-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 + 0.301·11-s + 0.288·12-s − 1.94·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s + 1.66·23-s + 0.204·24-s − 4/5·25-s − 1.37·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.60684636702857, −14.01069066642384, −13.69893519850819, −13.29654612590845, −12.65603949846429, −12.23396945326298, −11.70648887851542, −11.29007755434487, −10.31269708190852, −10.11190454752898, −9.512667115929413, −9.070910820281223, −8.453805643182942, −7.695522627122733, −7.176187707127336, −6.782184388809361, −6.297184889549026, −5.349591992957694, −4.932430151853838, −4.613083771681361, −3.571788544955472, −3.203948179099852, −2.493882032361108, −2.041503920330730, −1.167631067447509, 0,
1.167631067447509, 2.041503920330730, 2.493882032361108, 3.203948179099852, 3.571788544955472, 4.613083771681361, 4.932430151853838, 5.349591992957694, 6.297184889549026, 6.782184388809361, 7.176187707127336, 7.695522627122733, 8.453805643182942, 9.070910820281223, 9.512667115929413, 10.11190454752898, 10.31269708190852, 11.29007755434487, 11.70648887851542, 12.23396945326298, 12.65603949846429, 13.29654612590845, 13.69893519850819, 14.01069066642384, 14.60684636702857
