| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 11-s + 6·13-s + 2·15-s − 4·19-s − 4·21-s − 25-s − 27-s − 2·29-s − 33-s − 8·35-s + 10·37-s − 6·39-s + 10·41-s + 4·43-s − 2·45-s − 4·47-s + 9·49-s − 2·53-s − 2·55-s + 4·57-s − 2·61-s + 4·63-s − 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.917·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.174·33-s − 1.35·35-s + 1.64·37-s − 0.960·39-s + 1.56·41-s + 0.609·43-s − 0.298·45-s − 0.583·47-s + 9/7·49-s − 0.274·53-s − 0.269·55-s + 0.529·57-s − 0.256·61-s + 0.503·63-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 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.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.57289250696498, −13.05090568975636, −12.47101735304723, −12.09102540959063, −11.40727046410978, −11.25799621748308, −10.87277352761284, −10.60581987881638, −9.645916138830774, −9.218159561780343, −8.557063044241392, −8.189841634905395, −7.791533014507567, −7.392594867652095, −6.601716188713106, −6.120593426704358, −5.741058052721782, −5.044771450521321, −4.429357607301474, −4.080063921817182, −3.760720544690236, −2.802702058857885, −2.071259844110770, −1.335051549480543, −0.9711590103219861, 0,
0.9711590103219861, 1.335051549480543, 2.071259844110770, 2.802702058857885, 3.760720544690236, 4.080063921817182, 4.429357607301474, 5.044771450521321, 5.741058052721782, 6.120593426704358, 6.601716188713106, 7.392594867652095, 7.791533014507567, 8.189841634905395, 8.557063044241392, 9.218159561780343, 9.645916138830774, 10.60581987881638, 10.87277352761284, 11.25799621748308, 11.40727046410978, 12.09102540959063, 12.47101735304723, 13.05090568975636, 13.57289250696498
