| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 5·11-s + 12-s + 15-s + 16-s + 7·17-s − 18-s + 20-s − 5·22-s − 7·23-s − 24-s + 25-s + 27-s − 30-s + 10·31-s − 32-s + 5·33-s − 7·34-s + 36-s + 11·37-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 + 1.50·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 0.223·20-s − 1.06·22-s − 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.182·30-s + 1.79·31-s − 0.176·32-s + 0.870·33-s − 1.20·34-s + 1/6·36-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.268637551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.268637551\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.75867658498995, −12.26165417178417, −11.91426585534857, −11.56042460061274, −10.97564389313954, −10.24801575212683, −9.900205235670268, −9.695530157614823, −9.245301831668614, −8.587926295945721, −8.263402221666017, −7.739319648744020, −7.420912755148909, −6.598622965189286, −6.281523102885395, −5.967719509442056, −5.209025614367925, −4.551655195807703, −3.952108055305417, −3.513406870148606, −2.898478603972527, −2.316276869804778, −1.698698226304670, −1.122647971400325, −0.6887114031740275,
0.6887114031740275, 1.122647971400325, 1.698698226304670, 2.316276869804778, 2.898478603972527, 3.513406870148606, 3.952108055305417, 4.551655195807703, 5.209025614367925, 5.967719509442056, 6.281523102885395, 6.598622965189286, 7.420912755148909, 7.739319648744020, 8.263402221666017, 8.587926295945721, 9.245301831668614, 9.695530157614823, 9.900205235670268, 10.24801575212683, 10.97564389313954, 11.56042460061274, 11.91426585534857, 12.26165417178417, 12.75867658498995
