| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s − 15-s − 3·17-s − 19-s − 21-s − 6·23-s − 4·25-s − 5·27-s − 8·29-s − 8·31-s + 35-s + 5·37-s + 2·41-s + 43-s + 2·45-s + 3·47-s − 6·49-s − 3·51-s − 2·53-s − 57-s − 10·59-s − 14·61-s + 2·63-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.258·15-s − 0.727·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s − 0.962·27-s − 1.48·29-s − 1.43·31-s + 0.169·35-s + 0.821·37-s + 0.312·41-s + 0.152·43-s + 0.298·45-s + 0.437·47-s − 6/7·49-s − 0.420·51-s − 0.274·53-s − 0.132·57-s − 1.30·59-s − 1.79·61-s + 0.251·63-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−14.89537770206831, −14.55788636515792, −14.02818839897136, −13.40682154580440, −13.11696031578424, −12.47251683568638, −11.89265361876341, −11.42544446235779, −10.93292075284791, −10.43397506037165, −9.635760883365178, −9.105911474885297, −9.004358408919388, −7.977102798612947, −7.820026366467633, −7.297177202603007, −6.394464125865730, −5.968984622925735, −5.485680241386271, −4.551286753532744, −4.000218907800587, −3.519810335438287, −2.862012949799186, −2.140335593357610, −1.597651232320347, 0, 0,
1.597651232320347, 2.140335593357610, 2.862012949799186, 3.519810335438287, 4.000218907800587, 4.551286753532744, 5.485680241386271, 5.968984622925735, 6.394464125865730, 7.297177202603007, 7.820026366467633, 7.977102798612947, 9.004358408919388, 9.105911474885297, 9.635760883365178, 10.43397506037165, 10.93292075284791, 11.42544446235779, 11.89265361876341, 12.47251683568638, 13.11696031578424, 13.40682154580440, 14.02818839897136, 14.55788636515792, 14.89537770206831
