| L(s) = 1 | − 3·3-s + 3·5-s + 6·9-s − 3·11-s − 9·15-s + 2·17-s + 19-s + 4·25-s − 9·27-s + 7·29-s − 3·31-s + 9·33-s − 2·37-s + 3·41-s + 7·43-s + 18·45-s − 47-s − 6·51-s + 3·53-s − 9·55-s − 3·57-s + 4·59-s + 13·61-s − 3·67-s + 13·71-s − 13·73-s − 12·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 2·9-s − 0.904·11-s − 2.32·15-s + 0.485·17-s + 0.229·19-s + 4/5·25-s − 1.73·27-s + 1.29·29-s − 0.538·31-s + 1.56·33-s − 0.328·37-s + 0.468·41-s + 1.06·43-s + 2.68·45-s − 0.145·47-s − 0.840·51-s + 0.412·53-s − 1.21·55-s − 0.397·57-s + 0.520·59-s + 1.66·61-s − 0.366·67-s + 1.54·71-s − 1.52·73-s − 1.38·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.67415329506675, −13.01836013682933, −12.72762959861257, −12.34576546313944, −11.69947823300260, −11.33267635704611, −10.72202418545057, −10.32868191513883, −10.07362633485926, −9.594420482263213, −8.977017541443931, −8.340261428891217, −7.605751959504631, −7.181179172621935, −6.537289327565536, −6.164383016463157, −5.693151369596494, −5.213235084986477, −5.025879399243432, −4.266593061450212, −3.570521155998441, −2.611688147771975, −2.225970051208314, −1.307462749408461, −0.8927924909422719, 0,
0.8927924909422719, 1.307462749408461, 2.225970051208314, 2.611688147771975, 3.570521155998441, 4.266593061450212, 5.025879399243432, 5.213235084986477, 5.693151369596494, 6.164383016463157, 6.537289327565536, 7.181179172621935, 7.605751959504631, 8.340261428891217, 8.977017541443931, 9.594420482263213, 10.07362633485926, 10.32868191513883, 10.72202418545057, 11.33267635704611, 11.69947823300260, 12.34576546313944, 12.72762959861257, 13.01836013682933, 13.67415329506675
