| L(s) = 1 | − 2-s + 3-s − 4-s + 4·5-s − 6-s + 3·8-s − 2·9-s − 4·10-s + 11-s − 12-s + 4·15-s − 16-s + 3·17-s + 2·18-s − 8·19-s − 4·20-s − 22-s + 6·23-s + 3·24-s + 11·25-s − 5·27-s − 6·29-s − 4·30-s − 4·31-s − 5·32-s + 33-s − 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 1.78·5-s − 0.408·6-s + 1.06·8-s − 2/3·9-s − 1.26·10-s + 0.301·11-s − 0.288·12-s + 1.03·15-s − 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.83·19-s − 0.894·20-s − 0.213·22-s + 1.25·23-s + 0.612·24-s + 11/5·25-s − 0.962·27-s − 1.11·29-s − 0.730·30-s − 0.718·31-s − 0.883·32-s + 0.174·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91091 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.93882057993853, −13.76339043459345, −13.16513463599883, −12.81845399890030, −12.39567971132009, −11.39700705653546, −10.90678989579202, −10.47238952379052, −10.02763283657725, −9.470017890935516, −9.092722687147843, −8.720282194064866, −8.449997023711337, −7.577085044686508, −7.148715412390632, −6.407367826849136, −5.909033196874179, −5.391203437194995, −4.945270311817215, −4.159321521680390, −3.495982441577200, −2.811549121715912, −2.077305821979959, −1.751877948530224, −0.9738326022399251, 0,
0.9738326022399251, 1.751877948530224, 2.077305821979959, 2.811549121715912, 3.495982441577200, 4.159321521680390, 4.945270311817215, 5.391203437194995, 5.909033196874179, 6.407367826849136, 7.148715412390632, 7.577085044686508, 8.449997023711337, 8.720282194064866, 9.092722687147843, 9.470017890935516, 10.02763283657725, 10.47238952379052, 10.90678989579202, 11.39700705653546, 12.39567971132009, 12.81845399890030, 13.16513463599883, 13.76339043459345, 13.93882057993853
