| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 5·11-s + 12-s + 14-s − 15-s + 16-s + 17-s + 18-s + 6·19-s − 20-s + 21-s + 5·22-s − 4·23-s + 24-s − 4·25-s + 27-s + 28-s + 2·29-s − 30-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 + 1.50·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.927632801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.927632801\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 3331 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.45590778166913, −12.00316054091739, −11.86036471937067, −11.40414091059045, −10.93990410293275, −10.25807792938264, −9.801314837073154, −9.378777814618474, −9.023866688744390, −8.207004966273374, −7.955799616981649, −7.622790306005430, −6.912353355806689, −6.565421738904029, −6.043058398249893, −5.542760411406043, −4.799303990433952, −4.535420254792438, −3.905518297553126, −3.466558977849098, −3.198877173062681, −2.336237548119936, −1.832262188489118, −1.232430656088095, −0.6513284878701920,
0.6513284878701920, 1.232430656088095, 1.832262188489118, 2.336237548119936, 3.198877173062681, 3.466558977849098, 3.905518297553126, 4.535420254792438, 4.799303990433952, 5.542760411406043, 6.043058398249893, 6.565421738904029, 6.912353355806689, 7.622790306005430, 7.955799616981649, 8.207004966273374, 9.023866688744390, 9.378777814618474, 9.801314837073154, 10.25807792938264, 10.93990410293275, 11.40414091059045, 11.86036471937067, 12.00316054091739, 12.45590778166913
