| L(s) = 1 | − 2·2-s + 2·4-s + 3·11-s − 13-s − 4·16-s + 7·17-s − 6·22-s − 6·23-s + 2·26-s + 5·29-s − 2·31-s + 8·32-s − 14·34-s + 2·37-s + 2·41-s − 4·43-s + 6·44-s + 12·46-s − 3·47-s − 2·52-s − 6·53-s − 10·58-s + 10·59-s + 8·61-s + 4·62-s − 8·64-s + 2·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.904·11-s − 0.277·13-s − 16-s + 1.69·17-s − 1.27·22-s − 1.25·23-s + 0.392·26-s + 0.928·29-s − 0.359·31-s + 1.41·32-s − 2.40·34-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.904·44-s + 1.76·46-s − 0.437·47-s − 0.277·52-s − 0.824·53-s − 1.31·58-s + 1.30·59-s + 1.02·61-s + 0.508·62-s − 64-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024507914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024507914\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−16.49580860448651, −16.26264015322394, −15.64924679981491, −14.74009800037708, −14.28999103623705, −13.85856288110359, −12.91468865285357, −12.31532179249206, −11.65043923554442, −11.31493493790368, −10.30857150255935, −9.986431171269633, −9.593451662664112, −8.813330076431916, −8.261513933890406, −7.761986021829912, −7.126228819079624, −6.455724469428777, −5.752529403374460, −4.873374895633899, −4.041494887861638, −3.269258206189447, −2.215309395090345, −1.416331556885085, −0.6378419223994604,
0.6378419223994604, 1.416331556885085, 2.215309395090345, 3.269258206189447, 4.041494887861638, 4.873374895633899, 5.752529403374460, 6.455724469428777, 7.126228819079624, 7.761986021829912, 8.261513933890406, 8.813330076431916, 9.593451662664112, 9.986431171269633, 10.30857150255935, 11.31493493790368, 11.65043923554442, 12.31532179249206, 12.91468865285357, 13.85856288110359, 14.28999103623705, 14.74009800037708, 15.64924679981491, 16.26264015322394, 16.49580860448651
