| L(s) = 1 | − 2·2-s + 2·4-s + 4·7-s − 2·13-s − 8·14-s − 4·16-s − 5·17-s − 2·23-s + 4·26-s + 8·28-s − 5·29-s + 8·32-s + 10·34-s + 3·37-s + 41-s + 7·43-s + 4·46-s − 4·47-s + 9·49-s − 4·52-s − 53-s + 10·58-s + 12·59-s + 5·61-s − 8·64-s − 10·67-s − 10·68-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.51·7-s − 0.554·13-s − 2.13·14-s − 16-s − 1.21·17-s − 0.417·23-s + 0.784·26-s + 1.51·28-s − 0.928·29-s + 1.41·32-s + 1.71·34-s + 0.493·37-s + 0.156·41-s + 1.06·43-s + 0.589·46-s − 0.583·47-s + 9/7·49-s − 0.554·52-s − 0.137·53-s + 1.31·58-s + 1.56·59-s + 0.640·61-s − 64-s − 1.22·67-s − 1.21·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−7.46413710855759629641456624654, −7.15731968129842280205498903516, −6.19132109672873018026744637631, −5.27940067928201879543049041836, −4.58448918476243485796354308609, −4.00532331751181736905727109175, −2.50521474506863085036946073738, −1.97804162429170638663545196741, −1.14733685879658396316631001404, 0,
1.14733685879658396316631001404, 1.97804162429170638663545196741, 2.50521474506863085036946073738, 4.00532331751181736905727109175, 4.58448918476243485796354308609, 5.27940067928201879543049041836, 6.19132109672873018026744637631, 7.15731968129842280205498903516, 7.46413710855759629641456624654
