| L(s) = 1 | − 3-s − 5-s + 4·7-s − 2·9-s − 11-s + 2·13-s + 15-s + 2·19-s − 4·21-s + 9·23-s − 4·25-s + 5·27-s − 4·29-s + 5·31-s + 33-s − 4·35-s + 9·37-s − 2·39-s + 2·41-s + 6·43-s + 2·45-s − 4·47-s + 9·49-s + 6·53-s + 55-s − 2·57-s + 5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 0.458·19-s − 0.872·21-s + 1.87·23-s − 4/5·25-s + 0.962·27-s − 0.742·29-s + 0.898·31-s + 0.174·33-s − 0.676·35-s + 1.47·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s + 0.298·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 0.264·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300668157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300668157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−10.91267338620554434789681350953, −9.573240399409499405555120644843, −8.496357176715414070552880550880, −7.967328154488951241102396790972, −6.99641213903040183180071310948, −5.74481494741851233338307801250, −5.10113154745489541398926596759, −4.11605312880612782588264755648, −2.67932075932635757436168117174, −1.05195707157375879454007642879,
1.05195707157375879454007642879, 2.67932075932635757436168117174, 4.11605312880612782588264755648, 5.10113154745489541398926596759, 5.74481494741851233338307801250, 6.99641213903040183180071310948, 7.967328154488951241102396790972, 8.496357176715414070552880550880, 9.573240399409499405555120644843, 10.91267338620554434789681350953
