| L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·11-s + 4·13-s + 17-s − 19-s − 2·21-s + 3·23-s − 27-s − 8·29-s − 4·31-s + 2·33-s − 37-s − 4·39-s − 7·41-s − 8·47-s − 3·49-s − 51-s + 11·53-s + 57-s + 5·59-s + 13·61-s + 2·63-s + 8·67-s − 3·69-s − 15·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.242·17-s − 0.229·19-s − 0.436·21-s + 0.625·23-s − 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.348·33-s − 0.164·37-s − 0.640·39-s − 1.09·41-s − 1.16·47-s − 3/7·49-s − 0.140·51-s + 1.51·53-s + 0.132·57-s + 0.650·59-s + 1.66·61-s + 0.251·63-s + 0.977·67-s − 0.361·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.299513153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299513153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−12.43781371491838, −11.85526458033085, −11.51408552588953, −11.08473866389160, −10.82523686970330, −10.30671992271809, −9.820786854282405, −9.358663386977052, −8.592878027597898, −8.496177476154415, −7.942461064742829, −7.336238211421258, −6.965440120529562, −6.521011157286285, −5.758365910657240, −5.461104198480416, −5.200848443983275, −4.469647495530220, −3.994235227807740, −3.511068821274760, −2.916136944011968, −2.154900223094591, −1.600326286638660, −1.188635049399101, −0.3115326830454520,
0.3115326830454520, 1.188635049399101, 1.600326286638660, 2.154900223094591, 2.916136944011968, 3.511068821274760, 3.994235227807740, 4.469647495530220, 5.200848443983275, 5.461104198480416, 5.758365910657240, 6.521011157286285, 6.965440120529562, 7.336238211421258, 7.942461064742829, 8.496177476154415, 8.592878027597898, 9.358663386977052, 9.820786854282405, 10.30671992271809, 10.82523686970330, 11.08473866389160, 11.51408552588953, 11.85526458033085, 12.43781371491838
