| L(s) = 1 | + 2-s + 4-s + 2·5-s − 4·7-s + 8-s + 2·10-s + 2·11-s + 4·13-s − 4·14-s + 16-s + 2·17-s + 6·19-s + 2·20-s + 2·22-s + 8·23-s − 25-s + 4·26-s − 4·28-s − 8·31-s + 32-s + 2·34-s − 8·35-s + 2·37-s + 6·38-s + 2·40-s + 41-s − 12·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.51·7-s + 0.353·8-s + 0.632·10-s + 0.603·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.426·22-s + 1.66·23-s − 1/5·25-s + 0.784·26-s − 0.755·28-s − 1.43·31-s + 0.176·32-s + 0.342·34-s − 1.35·35-s + 0.328·37-s + 0.973·38-s + 0.316·40-s + 0.156·41-s − 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.577333595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.577333595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−10.34260298344213032152980942372, −9.500765621503071181245320047111, −9.002186278155841242025555524183, −7.47492130070894340231163417394, −6.56854224055727100168191315030, −5.98641655990333495752687077241, −5.13557219215566817531475310568, −3.60055397148664613714283361358, −3.07034922341211271188481825839, −1.41965243749365848176882424397,
1.41965243749365848176882424397, 3.07034922341211271188481825839, 3.60055397148664613714283361358, 5.13557219215566817531475310568, 5.98641655990333495752687077241, 6.56854224055727100168191315030, 7.47492130070894340231163417394, 9.002186278155841242025555524183, 9.500765621503071181245320047111, 10.34260298344213032152980942372
