| L(s) = 1 | − 2·4-s + 5·7-s − 5·13-s + 4·16-s − 8·19-s − 5·25-s − 10·28-s + 4·31-s + 11·37-s − 8·43-s + 18·49-s + 10·52-s + 14·61-s − 8·64-s + 16·67-s + 10·73-s + 16·76-s − 4·79-s − 25·91-s − 19·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 20·112-s + 113-s + ⋯ |
| L(s) = 1 | − 4-s + 1.88·7-s − 1.38·13-s + 16-s − 1.83·19-s − 25-s − 1.88·28-s + 0.718·31-s + 1.80·37-s − 1.21·43-s + 18/7·49-s + 1.38·52-s + 1.79·61-s − 64-s + 1.95·67-s + 1.17·73-s + 1.83·76-s − 0.450·79-s − 2.62·91-s − 1.92·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 1.88·112-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610739849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610739849\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 47 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−14.35884347303176, −14.00572566430552, −13.24742291759389, −12.91290621540236, −12.28837264027553, −11.76815941737316, −11.32922071874223, −10.76970564697837, −10.13932671438781, −9.750719481865503, −9.164458608897532, −8.422284616593507, −8.108130267873159, −7.894571953376765, −7.065929357620058, −6.424094997394420, −5.582431963720897, −5.172578662794121, −4.600919695770098, −4.283809221583931, −3.719102442863682, −2.471184635261530, −2.185507580665264, −1.308443956039197, −0.4580132354873869,
0.4580132354873869, 1.308443956039197, 2.185507580665264, 2.471184635261530, 3.719102442863682, 4.283809221583931, 4.600919695770098, 5.172578662794121, 5.582431963720897, 6.424094997394420, 7.065929357620058, 7.894571953376765, 8.108130267873159, 8.422284616593507, 9.164458608897532, 9.750719481865503, 10.13932671438781, 10.76970564697837, 11.32922071874223, 11.76815941737316, 12.28837264027553, 12.91290621540236, 13.24742291759389, 14.00572566430552, 14.35884347303176
