| L(s) = 1 | − 4·5-s − 2·11-s − 2·17-s − 8·19-s + 4·23-s + 11·25-s − 6·29-s − 4·31-s + 6·37-s − 12·41-s + 4·43-s + 6·47-s − 7·49-s − 2·53-s + 8·55-s − 14·59-s − 10·61-s + 4·67-s − 2·71-s + 2·73-s + 8·79-s + 14·83-s + 8·85-s + 32·95-s + 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.603·11-s − 0.485·17-s − 1.83·19-s + 0.834·23-s + 11/5·25-s − 1.11·29-s − 0.718·31-s + 0.986·37-s − 1.87·41-s + 0.609·43-s + 0.875·47-s − 49-s − 0.274·53-s + 1.07·55-s − 1.82·59-s − 1.28·61-s + 0.488·67-s − 0.237·71-s + 0.234·73-s + 0.900·79-s + 1.53·83-s + 0.867·85-s + 3.28·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−14.09763516764436, −13.34936378089457, −12.95484704796313, −12.55491688106419, −12.13711684118054, −11.47160868941173, −11.07077309296641, −10.75784498149749, −10.34971371112838, −9.360464833326296, −9.009108737107511, −8.448263422968824, −8.008153672154610, −7.520943511576922, −7.153307896266051, −6.440130782675676, −6.033866358680110, −5.003521974935740, −4.751932508103827, −4.155673133825367, −3.593156430530488, −3.134279150308727, −2.349141071390988, −1.663893824747726, −0.5539496694006162, 0,
0.5539496694006162, 1.663893824747726, 2.349141071390988, 3.134279150308727, 3.593156430530488, 4.155673133825367, 4.751932508103827, 5.003521974935740, 6.033866358680110, 6.440130782675676, 7.153307896266051, 7.520943511576922, 8.008153672154610, 8.448263422968824, 9.009108737107511, 9.360464833326296, 10.34971371112838, 10.75784498149749, 11.07077309296641, 11.47160868941173, 12.13711684118054, 12.55491688106419, 12.95484704796313, 13.34936378089457, 14.09763516764436
