| L(s) = 1 | − 3-s − 2·4-s + 5-s + 9-s + 2·12-s + 13-s − 15-s + 4·16-s − 6·17-s − 5·19-s − 2·20-s + 6·23-s + 25-s − 27-s − 6·29-s − 5·31-s − 2·36-s − 7·37-s − 39-s − 12·41-s − 43-s + 45-s − 6·47-s − 4·48-s + 6·51-s − 2·52-s + 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s + 0.577·12-s + 0.277·13-s − 0.258·15-s + 16-s − 1.45·17-s − 1.14·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s − 1/3·36-s − 1.15·37-s − 0.160·39-s − 1.87·41-s − 0.152·43-s + 0.149·45-s − 0.875·47-s − 0.577·48-s + 0.840·51-s − 0.277·52-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−9.995796123322737352351474895831, −8.960945268524742896229550337862, −8.603472667172365262215880298161, −7.18962953853699406184660490974, −6.34498577185094261386989897294, −5.32128135563663028070385926165, −4.61470299997904821875511065104, −3.53725365281124946738454390453, −1.80163328686849374891334706368, 0,
1.80163328686849374891334706368, 3.53725365281124946738454390453, 4.61470299997904821875511065104, 5.32128135563663028070385926165, 6.34498577185094261386989897294, 7.18962953853699406184660490974, 8.603472667172365262215880298161, 8.960945268524742896229550337862, 9.995796123322737352351474895831
