| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s + 2·7-s − 8-s − 2·9-s − 3·10-s − 3·11-s − 12-s − 2·13-s − 2·14-s − 3·15-s + 16-s + 2·18-s + 3·20-s − 2·21-s + 3·22-s + 23-s + 24-s + 4·25-s + 2·26-s + 5·27-s + 2·28-s + 3·29-s + 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.904·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.471·18-s + 0.670·20-s − 0.436·21-s + 0.639·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.962·27-s + 0.377·28-s + 0.557·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16606 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16606 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−16.62546887243319, −15.64877449103059, −15.15078056702886, −14.56893682963185, −13.89564240323203, −13.57634461220494, −12.79935506151184, −12.17752237175578, −11.57593332011282, −11.10029422978687, −10.36800538678750, −10.16450300292292, −9.457600533690560, −8.846206734231121, −8.127960558437572, −7.816513911746705, −6.727877010863940, −6.406206116932207, −5.628514966881602, −5.110001240627924, −4.721653234099573, −3.265214128892579, −2.546710609763996, −1.968290477594983, −1.094460745298878, 0,
1.094460745298878, 1.968290477594983, 2.546710609763996, 3.265214128892579, 4.721653234099573, 5.110001240627924, 5.628514966881602, 6.406206116932207, 6.727877010863940, 7.816513911746705, 8.127960558437572, 8.846206734231121, 9.457600533690560, 10.16450300292292, 10.36800538678750, 11.10029422978687, 11.57593332011282, 12.17752237175578, 12.79935506151184, 13.57634461220494, 13.89564240323203, 14.56893682963185, 15.15078056702886, 15.64877449103059, 16.62546887243319
