| L(s) = 1 | − 4·5-s + 2·11-s + 6·13-s − 2·17-s + 4·19-s − 23-s + 11·25-s − 4·29-s + 9·31-s + 8·37-s − 3·41-s − 2·43-s − 9·47-s + 12·53-s − 8·55-s + 4·59-s + 6·61-s − 24·65-s + 14·67-s − 71-s + 7·73-s + 3·79-s + 14·83-s + 8·85-s + 3·89-s − 16·95-s + 10·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s + 1.66·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 11/5·25-s − 0.742·29-s + 1.61·31-s + 1.31·37-s − 0.468·41-s − 0.304·43-s − 1.31·47-s + 1.64·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s − 2.97·65-s + 1.71·67-s − 0.118·71-s + 0.819·73-s + 0.337·79-s + 1.53·83-s + 0.867·85-s + 0.317·89-s − 1.64·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.276534136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.276534136\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−14.34495344597019, −13.63449693189390, −13.27697809275441, −12.75119238275919, −11.96079433221815, −11.71027798825833, −11.33075414506393, −10.95234164603689, −10.25101473379777, −9.608696613006385, −8.976868674484977, −8.450511197596438, −8.081605609582599, −7.676949299959417, −6.820159824948405, −6.616831705845542, −5.890626236836556, −5.103690737238562, −4.503041810784970, −3.857825559039191, −3.629274080379460, −3.009557795894046, −2.054058336687594, −0.9878874220826673, −0.6676682381497971,
0.6676682381497971, 0.9878874220826673, 2.054058336687594, 3.009557795894046, 3.629274080379460, 3.857825559039191, 4.503041810784970, 5.103690737238562, 5.890626236836556, 6.616831705845542, 6.820159824948405, 7.676949299959417, 8.081605609582599, 8.450511197596438, 8.976868674484977, 9.608696613006385, 10.25101473379777, 10.95234164603689, 11.33075414506393, 11.71027798825833, 11.96079433221815, 12.75119238275919, 13.27697809275441, 13.63449693189390, 14.34495344597019
