| L(s) = 1 | + 3-s − 2·7-s + 9-s − 4·13-s + 2·17-s − 2·21-s + 2·23-s − 5·25-s + 27-s + 2·29-s + 4·31-s + 6·37-s − 4·39-s + 6·41-s − 12·43-s + 6·47-s − 3·49-s + 2·51-s − 2·63-s − 4·67-s + 2·69-s + 10·71-s − 2·73-s − 5·75-s + 2·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.485·17-s − 0.436·21-s + 0.417·23-s − 25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.640·39-s + 0.937·41-s − 1.82·43-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.251·63-s − 0.488·67-s + 0.240·69-s + 1.18·71-s − 0.234·73-s − 0.577·75-s + 0.225·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−16.76756653811750, −16.06518386807363, −15.52470135335677, −15.00879170055742, −14.44415490927525, −13.90110915394402, −13.26380217533552, −12.81282707332928, −12.13755337675725, −11.71418820268166, −10.84839093812922, −10.10734619783385, −9.674733692041705, −9.316320273905396, −8.383826010626848, −7.895074844449061, −7.230372879276636, −6.628424951012084, −5.925729276770750, −5.127149145927394, −4.401345279523217, −3.649716824705062, −2.875081145919316, −2.350445895858594, −1.218941462429640, 0,
1.218941462429640, 2.350445895858594, 2.875081145919316, 3.649716824705062, 4.401345279523217, 5.127149145927394, 5.925729276770750, 6.628424951012084, 7.230372879276636, 7.895074844449061, 8.383826010626848, 9.316320273905396, 9.674733692041705, 10.10734619783385, 10.84839093812922, 11.71418820268166, 12.13755337675725, 12.81282707332928, 13.26380217533552, 13.90110915394402, 14.44415490927525, 15.00879170055742, 15.52470135335677, 16.06518386807363, 16.76756653811750
