| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 3·11-s − 5·13-s + 16-s + 17-s − 3·19-s + 20-s + 3·22-s − 5·23-s + 25-s + 5·26-s − 2·31-s − 32-s − 34-s + 7·37-s + 3·38-s − 40-s − 9·41-s + 2·43-s − 3·44-s + 5·46-s + 47-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.904·11-s − 1.38·13-s + 1/4·16-s + 0.242·17-s − 0.688·19-s + 0.223·20-s + 0.639·22-s − 1.04·23-s + 1/5·25-s + 0.980·26-s − 0.359·31-s − 0.176·32-s − 0.171·34-s + 1.15·37-s + 0.486·38-s − 0.158·40-s − 1.40·41-s + 0.304·43-s − 0.452·44-s + 0.737·46-s + 0.145·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4761108265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4761108265\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.25563361725398, −13.41131069609413, −13.11568938124919, −12.49496237471149, −12.03147197386251, −11.60899513735662, −10.85911115344210, −10.43186456728078, −9.983277404591843, −9.674151651375552, −9.034613285747323, −8.440942328639444, −7.966476939347071, −7.443052282494269, −7.031337941573406, −6.289416015373825, −5.822741527500749, −5.212852097169090, −4.677898696374207, −3.975210063235883, −3.124353550806578, −2.450838425137442, −2.142576960156277, −1.306585892838356, −0.2498044260000997,
0.2498044260000997, 1.306585892838356, 2.142576960156277, 2.450838425137442, 3.124353550806578, 3.975210063235883, 4.677898696374207, 5.212852097169090, 5.822741527500749, 6.289416015373825, 7.031337941573406, 7.443052282494269, 7.966476939347071, 8.440942328639444, 9.034613285747323, 9.674151651375552, 9.983277404591843, 10.43186456728078, 10.85911115344210, 11.60899513735662, 12.03147197386251, 12.49496237471149, 13.11568938124919, 13.41131069609413, 14.25563361725398
