| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 3·9-s + 10-s − 11-s + 2·13-s − 16-s + 6·17-s + 3·18-s + 4·19-s + 20-s + 22-s − 4·23-s + 25-s − 2·26-s + 6·29-s + 8·31-s − 5·32-s − 6·34-s + 3·36-s − 2·37-s − 4·38-s − 3·40-s − 2·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 9-s + 0.316·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 1.11·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s − 0.328·37-s − 0.648·38-s − 0.474·40-s − 0.312·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 53 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 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.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.80647272004430, −12.90155871868874, −12.76547300947268, −11.88283677328481, −11.55711066413275, −11.38301153730857, −10.36396576254593, −10.12582585945074, −9.918795069949134, −9.092690579690367, −8.680782285660454, −8.225264089290038, −7.879355648365840, −7.561020363334741, −6.719448238538996, −6.200545915427498, −5.614318350109299, −5.038087199979016, −4.717094483731650, −3.869242682269614, −3.378998854505666, −2.951872188881792, −2.112949809557826, −1.202614000849567, −0.8054597165942609, 0,
0.8054597165942609, 1.202614000849567, 2.112949809557826, 2.951872188881792, 3.378998854505666, 3.869242682269614, 4.717094483731650, 5.038087199979016, 5.614318350109299, 6.200545915427498, 6.719448238538996, 7.561020363334741, 7.879355648365840, 8.225264089290038, 8.680782285660454, 9.092690579690367, 9.918795069949134, 10.12582585945074, 10.36396576254593, 11.38301153730857, 11.55711066413275, 11.88283677328481, 12.76547300947268, 12.90155871868874, 13.80647272004430
