| L(s) = 1 | + 7-s + 2·11-s + 4·13-s + 2·17-s + 2·19-s + 4·23-s − 2·29-s − 6·31-s − 6·37-s − 6·41-s − 4·43-s + 49-s − 8·53-s + 10·61-s − 12·67-s + 14·71-s − 4·73-s + 2·77-s − 8·79-s − 12·83-s + 14·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s + 1.10·13-s + 0.485·17-s + 0.458·19-s + 0.834·23-s − 0.371·29-s − 1.07·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 1.09·53-s + 1.28·61-s − 1.46·67-s + 1.66·71-s − 0.468·73-s + 0.227·77-s − 0.900·79-s − 1.31·83-s + 1.48·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.84506273919578, −13.69761863007318, −12.88714606269208, −12.68964305771045, −11.90042873197603, −11.54667377473483, −11.09799541901591, −10.65738220380719, −10.02728070802722, −9.556952720687335, −8.901008840059341, −8.623308420536532, −8.088287256817426, −7.350987190403663, −7.050491194835895, −6.378114562830080, −5.858503190734998, −5.268713409557917, −4.843008474561465, −4.061892501572589, −3.454518510069674, −3.217628283577021, −2.165693960098305, −1.504027859328394, −1.076601251048458, 0,
1.076601251048458, 1.504027859328394, 2.165693960098305, 3.217628283577021, 3.454518510069674, 4.061892501572589, 4.843008474561465, 5.268713409557917, 5.858503190734998, 6.378114562830080, 7.050491194835895, 7.350987190403663, 8.088287256817426, 8.623308420536532, 8.901008840059341, 9.556952720687335, 10.02728070802722, 10.65738220380719, 11.09799541901591, 11.54667377473483, 11.90042873197603, 12.68964305771045, 12.88714606269208, 13.69761863007318, 13.84506273919578
