| L(s) = 1 | − 2·4-s + 2·7-s − 3·11-s − 2·13-s + 4·16-s − 7·19-s − 6·23-s − 4·28-s − 3·29-s − 8·31-s − 4·37-s + 9·41-s − 2·43-s + 6·44-s − 12·47-s − 3·49-s + 4·52-s + 9·59-s + 7·61-s − 8·64-s + 16·67-s − 3·71-s − 16·73-s + 14·76-s − 6·77-s + 79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s − 0.904·11-s − 0.554·13-s + 16-s − 1.60·19-s − 1.25·23-s − 0.755·28-s − 0.557·29-s − 1.43·31-s − 0.657·37-s + 1.40·41-s − 0.304·43-s + 0.904·44-s − 1.75·47-s − 3/7·49-s + 0.554·52-s + 1.17·59-s + 0.896·61-s − 64-s + 1.95·67-s − 0.356·71-s − 1.87·73-s + 1.60·76-s − 0.683·77-s + 0.112·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.62216750157495, −14.37829035215679, −13.75872420707518, −13.03664406616606, −12.79710448897968, −12.51765108729439, −11.54287810988024, −11.28868529928722, −10.55972341263788, −10.13148297333893, −9.718094128103992, −9.028281547196278, −8.495049849088326, −8.112081575809470, −7.684835829276868, −7.031513847466037, −6.289749639988953, −5.562769321357825, −5.249742220444892, −4.642735051017204, −4.056302287845489, −3.653175847598728, −2.631587789911877, −2.057794744091466, −1.375522930297209, 0, 0,
1.375522930297209, 2.057794744091466, 2.631587789911877, 3.653175847598728, 4.056302287845489, 4.642735051017204, 5.249742220444892, 5.562769321357825, 6.289749639988953, 7.031513847466037, 7.684835829276868, 8.112081575809470, 8.495049849088326, 9.028281547196278, 9.718094128103992, 10.13148297333893, 10.55972341263788, 11.28868529928722, 11.54287810988024, 12.51765108729439, 12.79710448897968, 13.03664406616606, 13.75872420707518, 14.37829035215679, 14.62216750157495
