| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s − 2·9-s − 12-s − 2·14-s + 16-s + 17-s + 2·18-s + 19-s − 2·21-s − 6·23-s + 24-s + 5·27-s + 2·28-s − 9·29-s + 31-s − 32-s − 34-s − 2·36-s − 4·37-s − 38-s + 6·41-s + 2·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 0.962·27-s + 0.377·28-s − 1.67·29-s + 0.179·31-s − 0.176·32-s − 0.171·34-s − 1/3·36-s − 0.657·37-s − 0.162·38-s + 0.937·41-s + 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8258496026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8258496026\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.41175920530321, −12.72206397898315, −12.28850178037770, −11.73466010427835, −11.46245941057199, −11.02262644246607, −10.58275304458362, −10.04713691938354, −9.509615418640198, −9.071172876589692, −8.465843249788819, −7.992911340111220, −7.710343307424486, −7.063701187088608, −6.407224045210368, −6.034184526671320, −5.386428253996382, −5.114319602505440, −4.355821331782488, −3.637609859060634, −3.165599733316968, −2.198194927983721, −1.928875314602355, −1.076215608960600, −0.3455236819010576,
0.3455236819010576, 1.076215608960600, 1.928875314602355, 2.198194927983721, 3.165599733316968, 3.637609859060634, 4.355821331782488, 5.114319602505440, 5.386428253996382, 6.034184526671320, 6.407224045210368, 7.063701187088608, 7.710343307424486, 7.992911340111220, 8.465843249788819, 9.071172876589692, 9.509615418640198, 10.04713691938354, 10.58275304458362, 11.02262644246607, 11.46245941057199, 11.73466010427835, 12.28850178037770, 12.72206397898315, 13.41175920530321
