| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s + 13-s + 15-s + 16-s + 6·17-s − 18-s + 20-s − 4·23-s − 24-s + 25-s − 26-s + 27-s − 10·29-s − 30-s − 32-s − 6·34-s + 36-s − 6·37-s + 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.223·20-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.85·29-s − 0.182·30-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.986·37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−16.10198830072135, −15.42516774270268, −14.96178951955221, −14.35768143679007, −13.90129360403642, −13.35128189793387, −12.59232149392846, −12.23236569434546, −11.48617793539639, −10.86928724904654, −10.29083604279062, −9.699129783694530, −9.399676548996446, −8.701364719917595, −8.007154269343661, −7.732606419425324, −6.942985993565964, −6.345744716779139, −5.567920695303564, −5.131409776171404, −3.908551158392335, −3.489286413187119, −2.654987692801939, −1.820691860587158, −1.290942851856955, 0,
1.290942851856955, 1.820691860587158, 2.654987692801939, 3.489286413187119, 3.908551158392335, 5.131409776171404, 5.567920695303564, 6.345744716779139, 6.942985993565964, 7.732606419425324, 8.007154269343661, 8.701364719917595, 9.399676548996446, 9.699129783694530, 10.29083604279062, 10.86928724904654, 11.48617793539639, 12.23236569434546, 12.59232149392846, 13.35128189793387, 13.90129360403642, 14.35768143679007, 14.96178951955221, 15.42516774270268, 16.10198830072135
