| L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 11-s − 6·13-s + 7·17-s − 5·19-s + 3·21-s + 6·23-s + 5·27-s − 5·29-s − 3·31-s + 33-s + 3·37-s + 6·39-s + 2·41-s + 4·43-s + 2·47-s + 2·49-s − 7·51-s − 53-s + 5·57-s + 10·59-s − 7·61-s + 6·63-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s − 1.14·19-s + 0.654·21-s + 1.25·23-s + 0.962·27-s − 0.928·29-s − 0.538·31-s + 0.174·33-s + 0.493·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.291·47-s + 2/7·49-s − 0.980·51-s − 0.137·53-s + 0.662·57-s + 1.30·59-s − 0.896·61-s + 0.755·63-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−16.36684128961547, −15.55952978489169, −14.91520159182080, −14.55661384144285, −14.05400981030000, −13.07854149408006, −12.65607411070528, −12.44544898793786, −11.67511976525680, −11.12789250184150, −10.46585273688584, −9.961434777751106, −9.444965013786979, −8.883313181348262, −8.049595678561631, −7.388395125465840, −6.935926442418321, −6.158850313397233, −5.597182150390908, −5.150450298037823, −4.348251827485692, −3.390646486245590, −2.874315025825923, −2.160288364082797, −0.7889478788010654, 0,
0.7889478788010654, 2.160288364082797, 2.874315025825923, 3.390646486245590, 4.348251827485692, 5.150450298037823, 5.597182150390908, 6.158850313397233, 6.935926442418321, 7.388395125465840, 8.049595678561631, 8.883313181348262, 9.444965013786979, 9.961434777751106, 10.46585273688584, 11.12789250184150, 11.67511976525680, 12.44544898793786, 12.65607411070528, 13.07854149408006, 14.05400981030000, 14.55661384144285, 14.91520159182080, 15.55952978489169, 16.36684128961547
