| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s − 2·9-s − 10-s − 12-s − 3·14-s − 15-s + 16-s + 3·17-s + 2·18-s + 3·19-s + 20-s − 3·21-s + 24-s + 25-s + 5·27-s + 3·28-s + 9·29-s + 30-s − 5·31-s − 32-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.688·19-s + 0.223·20-s − 0.654·21-s + 0.204·24-s + 1/5·25-s + 0.962·27-s + 0.566·28-s + 1.67·29-s + 0.182·30-s − 0.898·31-s − 0.176·32-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.148842116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148842116\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−9.867093594057512094736215356103, −8.759639092048162651010568859765, −8.295199311121096801342382850468, −7.34064528302597351543746248338, −6.45878379574846454971229715659, −5.44872347708205944646551864373, −4.98686971409463722448062023795, −3.39105689658683572292638145407, −2.13538284795201551094974408589, −0.949917669447758428883942777957,
0.949917669447758428883942777957, 2.13538284795201551094974408589, 3.39105689658683572292638145407, 4.98686971409463722448062023795, 5.44872347708205944646551864373, 6.45878379574846454971229715659, 7.34064528302597351543746248338, 8.295199311121096801342382850468, 8.759639092048162651010568859765, 9.867093594057512094736215356103
