| L(s) = 1 | + 2·7-s − 11-s − 6·13-s − 4·17-s + 2·19-s + 8·23-s + 6·37-s + 10·43-s − 3·49-s + 14·53-s − 12·59-s − 14·61-s + 4·67-s − 6·73-s − 2·77-s − 2·79-s − 16·83-s + 14·89-s − 12·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s − 1.66·13-s − 0.970·17-s + 0.458·19-s + 1.66·23-s + 0.986·37-s + 1.52·43-s − 3/7·49-s + 1.92·53-s − 1.56·59-s − 1.79·61-s + 0.488·67-s − 0.702·73-s − 0.227·77-s − 0.225·79-s − 1.75·83-s + 1.48·89-s − 1.25·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.01733724726700, −14.62454144540875, −14.11208132507327, −13.41886317391235, −13.06527008902383, −12.36553473292296, −11.99695624360975, −11.34407067293953, −10.86783878754921, −10.47011753213102, −9.674850938457952, −9.198101831007835, −8.827814222784940, −7.876865502561420, −7.650103190788838, −7.021864733800365, −6.517581711266822, −5.532487629274101, −5.223530425904808, −4.466341260520836, −4.236346133176533, −2.889490712538370, −2.724181405834627, −1.843581222694258, −0.9965088536222552, 0,
0.9965088536222552, 1.843581222694258, 2.724181405834627, 2.889490712538370, 4.236346133176533, 4.466341260520836, 5.223530425904808, 5.532487629274101, 6.517581711266822, 7.021864733800365, 7.650103190788838, 7.876865502561420, 8.827814222784940, 9.198101831007835, 9.674850938457952, 10.47011753213102, 10.86783878754921, 11.34407067293953, 11.99695624360975, 12.36553473292296, 13.06527008902383, 13.41886317391235, 14.11208132507327, 14.62454144540875, 15.01733724726700
