| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s + 4·7-s − 8-s + 9-s − 10-s + 2·11-s + 2·12-s − 4·14-s + 2·15-s + 16-s + 2·17-s − 18-s − 6·19-s + 20-s + 8·21-s − 2·22-s + 6·23-s − 2·24-s + 25-s − 4·27-s + 4·28-s + 2·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.577·12-s − 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s + 1.74·21-s − 0.426·22-s + 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.769·27-s + 0.755·28-s + 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.458840227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458840227\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.025060619869158898794133991071, −8.585748447644376465948104393438, −8.100118890934652521098165508208, −7.23199397402297436337665710295, −6.34604314687718963427506007589, −5.20744001985508991082692307203, −4.27248516331024539785541305385, −3.06745890083762284356579944006, −2.12823848907769259336957975144, −1.31054759042860725094575746619,
1.31054759042860725094575746619, 2.12823848907769259336957975144, 3.06745890083762284356579944006, 4.27248516331024539785541305385, 5.20744001985508991082692307203, 6.34604314687718963427506007589, 7.23199397402297436337665710295, 8.100118890934652521098165508208, 8.585748447644376465948104393438, 9.025060619869158898794133991071
