| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 6·11-s − 6·13-s − 15-s − 4·17-s + 4·21-s − 6·23-s + 25-s − 27-s + 4·29-s − 31-s + 6·33-s − 4·35-s − 2·37-s + 6·39-s − 10·41-s − 12·43-s + 45-s − 4·47-s + 9·49-s + 4·51-s + 6·53-s − 6·55-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s − 0.258·15-s − 0.970·17-s + 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.179·31-s + 1.04·33-s − 0.676·35-s − 0.328·37-s + 0.960·39-s − 1.56·41-s − 1.82·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.809·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−6.93768187327729932799349069187, −6.62408648137275531097269565198, −5.77129354679514019587239448118, −5.12551341930854713882773931005, −4.58458190944135833533609792667, −3.38504372456866745778795052298, −2.65775420083474638129094215239, −1.98817992046031565267791338086, 0, 0,
1.98817992046031565267791338086, 2.65775420083474638129094215239, 3.38504372456866745778795052298, 4.58458190944135833533609792667, 5.12551341930854713882773931005, 5.77129354679514019587239448118, 6.62408648137275531097269565198, 6.93768187327729932799349069187
