| L(s) = 1 | − 2·4-s − 11-s − 5·13-s + 4·16-s + 4·17-s − 3·19-s + 8·23-s + 4·29-s + 31-s − 7·37-s + 4·41-s − 43-s + 2·44-s − 8·47-s + 10·52-s − 12·53-s − 2·61-s − 8·64-s − 3·67-s − 8·68-s + 4·71-s + 11·73-s + 6·76-s + 15·79-s + 12·83-s − 12·89-s − 16·92-s + ⋯ |
| L(s) = 1 | − 4-s − 0.301·11-s − 1.38·13-s + 16-s + 0.970·17-s − 0.688·19-s + 1.66·23-s + 0.742·29-s + 0.179·31-s − 1.15·37-s + 0.624·41-s − 0.152·43-s + 0.301·44-s − 1.16·47-s + 1.38·52-s − 1.64·53-s − 0.256·61-s − 64-s − 0.366·67-s − 0.970·68-s + 0.474·71-s + 1.28·73-s + 0.688·76-s + 1.68·79-s + 1.31·83-s − 1.27·89-s − 1.66·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−13.79984260159113, −13.35100510517492, −12.67779935937565, −12.45912382911301, −12.15544878673758, −11.32100277185538, −10.81843074993539, −10.31653563361756, −9.845531568168956, −9.412292777957112, −9.012141228729495, −8.351340040035748, −7.913556828123891, −7.529321737617388, −6.768822407853559, −6.411876017224073, −5.510344096141791, −5.061181677741118, −4.840732368094746, −4.205757583484264, −3.387180726915881, −3.053699254752119, −2.317459832347036, −1.473070903336654, −0.7340528438463978, 0,
0.7340528438463978, 1.473070903336654, 2.317459832347036, 3.053699254752119, 3.387180726915881, 4.205757583484264, 4.840732368094746, 5.061181677741118, 5.510344096141791, 6.411876017224073, 6.768822407853559, 7.529321737617388, 7.913556828123891, 8.351340040035748, 9.012141228729495, 9.412292777957112, 9.845531568168956, 10.31653563361756, 10.81843074993539, 11.32100277185538, 12.15544878673758, 12.45912382911301, 12.67779935937565, 13.35100510517492, 13.79984260159113
