| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 2·11-s + 13-s − 15-s − 17-s − 2·19-s + 2·21-s − 4·23-s + 25-s − 27-s − 2·29-s − 10·31-s − 2·33-s − 2·35-s − 6·37-s − 39-s + 2·41-s − 8·43-s + 45-s − 2·47-s − 3·49-s + 51-s − 6·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s − 0.458·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.79·31-s − 0.348·33-s − 0.338·35-s − 0.986·37-s − 0.160·39-s + 0.312·41-s − 1.21·43-s + 0.149·45-s − 0.291·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.42226888004718, −12.92954068771167, −12.52000355766511, −12.20441030235787, −11.55118873694493, −11.13005424678147, −10.70049597482144, −10.21445533093093, −9.716352554271250, −9.306001292022250, −8.905207159988709, −8.331788083064727, −7.686180733545145, −7.180024319627207, −6.543958752966544, −6.383441636341088, −5.815772198524670, −5.332708410739458, −4.757941556298426, −4.141936089492434, −3.565566284939090, −3.220412643164099, −2.295272257919415, −1.729143615699775, −1.278179445404534, 0, 0,
1.278179445404534, 1.729143615699775, 2.295272257919415, 3.220412643164099, 3.565566284939090, 4.141936089492434, 4.757941556298426, 5.332708410739458, 5.815772198524670, 6.383441636341088, 6.543958752966544, 7.180024319627207, 7.686180733545145, 8.331788083064727, 8.905207159988709, 9.306001292022250, 9.716352554271250, 10.21445533093093, 10.70049597482144, 11.13005424678147, 11.55118873694493, 12.20441030235787, 12.52000355766511, 12.92954068771167, 13.42226888004718
