| L(s) = 1 | − 2-s + 4-s − 2·5-s − 4·7-s − 8-s + 2·10-s − 6·13-s + 4·14-s + 16-s + 4·19-s − 2·20-s − 25-s + 6·26-s − 4·28-s − 10·29-s + 8·31-s − 32-s + 8·35-s − 4·37-s − 4·38-s + 2·40-s − 10·43-s − 10·47-s + 9·49-s + 50-s − 6·52-s + 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 0.353·8-s + 0.632·10-s − 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.917·19-s − 0.447·20-s − 1/5·25-s + 1.17·26-s − 0.755·28-s − 1.85·29-s + 1.43·31-s − 0.176·32-s + 1.35·35-s − 0.657·37-s − 0.648·38-s + 0.316·40-s − 1.52·43-s − 1.45·47-s + 9/7·49-s + 0.141·50-s − 0.832·52-s + 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172062 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172062 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.26562625253812, −12.95770268006963, −12.33989691278726, −11.91892366855339, −11.68432957150063, −11.15315265031078, −10.35715203444863, −10.00029366724782, −9.684145095390288, −9.295205792944605, −8.720114414980075, −8.019493315215247, −7.714416853708862, −7.151565405461667, −6.852175045229121, −6.322179570714477, −5.617298819447734, −5.136701847631316, −4.460805799443487, −3.792348241332575, −3.222864211120490, −2.932455156787071, −2.182994221430753, −1.453292361795759, −0.4202762662564253, 0,
0.4202762662564253, 1.453292361795759, 2.182994221430753, 2.932455156787071, 3.222864211120490, 3.792348241332575, 4.460805799443487, 5.136701847631316, 5.617298819447734, 6.322179570714477, 6.852175045229121, 7.151565405461667, 7.714416853708862, 8.019493315215247, 8.720114414980075, 9.295205792944605, 9.684145095390288, 10.00029366724782, 10.35715203444863, 11.15315265031078, 11.68432957150063, 11.91892366855339, 12.33989691278726, 12.95770268006963, 13.26562625253812
