| L(s) = 1 | + 3·11-s + 2·13-s + 3·17-s − 19-s − 6·23-s − 5·25-s − 6·29-s + 4·31-s + 4·37-s − 9·41-s + 43-s + 6·47-s − 12·53-s + 3·59-s + 8·61-s − 5·67-s − 12·71-s − 11·73-s − 4·79-s + 12·83-s − 6·89-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.904·11-s + 0.554·13-s + 0.727·17-s − 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.657·37-s − 1.40·41-s + 0.152·43-s + 0.875·47-s − 1.64·53-s + 0.390·59-s + 1.02·61-s − 0.610·67-s − 1.42·71-s − 1.28·73-s − 0.450·79-s + 1.31·83-s − 0.635·89-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.728615831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728615831\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−12.81332772318755, −12.32320558747851, −11.74258495431250, −11.66664223963361, −11.07501239941372, −10.45113247774141, −10.02597211602090, −9.633052660462632, −9.177020115027250, −8.550423942513235, −8.250437858819985, −7.591851961326073, −7.332966716025479, −6.517046116583520, −6.192281366979349, −5.767925500237654, −5.258157828095472, −4.473988170572201, −4.047924417608666, −3.647783088521521, −3.071548370812188, −2.337372284690573, −1.653501677538254, −1.306999618575897, −0.3564185031201115,
0.3564185031201115, 1.306999618575897, 1.653501677538254, 2.337372284690573, 3.071548370812188, 3.647783088521521, 4.047924417608666, 4.473988170572201, 5.258157828095472, 5.767925500237654, 6.192281366979349, 6.517046116583520, 7.332966716025479, 7.591851961326073, 8.250437858819985, 8.550423942513235, 9.177020115027250, 9.633052660462632, 10.02597211602090, 10.45113247774141, 11.07501239941372, 11.66664223963361, 11.74258495431250, 12.32320558747851, 12.81332772318755
