| L(s) = 1 | − 2-s − 4-s + 3·8-s + 6·13-s − 16-s − 6·17-s + 4·23-s − 6·26-s + 2·29-s − 8·31-s − 5·32-s + 6·34-s − 10·37-s − 2·41-s + 4·43-s − 4·46-s + 12·47-s − 7·49-s − 6·52-s + 6·53-s − 2·58-s − 12·59-s − 2·61-s + 8·62-s + 7·64-s − 4·67-s + 6·68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.834·23-s − 1.17·26-s + 0.371·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 0.589·46-s + 1.75·47-s − 49-s − 0.832·52-s + 0.824·53-s − 0.262·58-s − 1.56·59-s − 0.256·61-s + 1.01·62-s + 7/8·64-s − 0.488·67-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−14.12817893896867, −13.62550091085328, −13.28741817290003, −12.92170369358720, −12.26535311648162, −11.61678626469349, −10.97142746367935, −10.66082855649960, −10.44526746740743, −9.502882773150190, −8.956489941335211, −8.856137350419739, −8.430459372556465, −7.590943147299974, −7.264001957763132, −6.562158852799775, −6.064166906789131, −5.371167445557774, −4.810710372450953, −4.199550760204284, −3.692667011386289, −3.096224104500486, −2.073259455619370, −1.552960220458742, −0.8143782224268103, 0,
0.8143782224268103, 1.552960220458742, 2.073259455619370, 3.096224104500486, 3.692667011386289, 4.199550760204284, 4.810710372450953, 5.371167445557774, 6.064166906789131, 6.562158852799775, 7.264001957763132, 7.590943147299974, 8.430459372556465, 8.856137350419739, 8.956489941335211, 9.502882773150190, 10.44526746740743, 10.66082855649960, 10.97142746367935, 11.61678626469349, 12.26535311648162, 12.92170369358720, 13.28741817290003, 13.62550091085328, 14.12817893896867
