| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 11-s + 2·13-s − 15-s + 6·17-s + 4·19-s + 4·21-s − 8·23-s + 25-s − 27-s − 10·29-s + 33-s − 4·35-s − 10·37-s − 2·39-s − 6·41-s − 8·43-s + 45-s + 9·49-s − 6·51-s − 2·53-s − 55-s − 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.174·33-s − 0.676·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 9/7·49-s − 0.840·51-s − 0.274·53-s − 0.134·55-s − 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.606538228010066083296458195385, −8.454778415279917764375509620642, −7.42857632690023926592219584708, −6.66592679296296942964480595467, −5.75307988608423120622604654757, −5.41102754479565273282369074440, −3.80807868434124417046745598346, −3.18329722249466860720838736711, −1.64389654064481817822444150223, 0,
1.64389654064481817822444150223, 3.18329722249466860720838736711, 3.80807868434124417046745598346, 5.41102754479565273282369074440, 5.75307988608423120622604654757, 6.66592679296296942964480595467, 7.42857632690023926592219584708, 8.454778415279917764375509620642, 9.606538228010066083296458195385
