| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s − 4·13-s − 14-s − 15-s + 16-s − 18-s + 4·19-s − 20-s + 21-s + 6·23-s − 24-s + 25-s + 4·26-s + 27-s + 28-s + 2·29-s + 30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.10·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.996350969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.996350969\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.31334105955029, −15.02807978670407, −14.23104925485720, −13.98009616010906, −13.18760752210617, −12.53389006196143, −12.06473550456985, −11.52714111857173, −11.00523383571772, −10.28817770688772, −9.839007976005538, −9.294277960957835, −8.685949637711695, −8.225817470425458, −7.475715478174820, −7.329454993133178, −6.590800557333479, −5.786109862681687, −4.871148334082996, −4.588101681470615, −3.545148070304659, −2.885673476623592, −2.386143002225390, −1.367809315464899, −0.6471755359335562,
0.6471755359335562, 1.367809315464899, 2.386143002225390, 2.885673476623592, 3.545148070304659, 4.588101681470615, 4.871148334082996, 5.786109862681687, 6.590800557333479, 7.329454993133178, 7.475715478174820, 8.225817470425458, 8.685949637711695, 9.294277960957835, 9.839007976005538, 10.28817770688772, 11.00523383571772, 11.52714111857173, 12.06473550456985, 12.53389006196143, 13.18760752210617, 13.98009616010906, 14.23104925485720, 15.02807978670407, 15.31334105955029
