| L(s) = 1 | + 11-s − 6·13-s + 2·17-s + 4·19-s + 10·29-s − 6·37-s − 2·41-s + 4·43-s + 8·47-s − 7·49-s − 10·53-s − 4·59-s − 2·61-s − 4·67-s − 8·71-s − 2·73-s + 8·79-s + 12·83-s + 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.85·29-s − 0.986·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.949·71-s − 0.234·73-s + 0.900·79-s + 1.31·83-s + 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(\approx\) |
\(1.939549676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939549676\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.72265469674490, −14.22244408304544, −13.92325292679160, −13.31995557977405, −12.46386309720638, −12.13668846690863, −11.97129498345862, −11.09609320468096, −10.54601650233434, −9.965574263495164, −9.597756795260887, −9.038170942227119, −8.354205506209427, −7.736280590172995, −7.303171355703238, −6.742291263848008, −6.091494594150493, −5.405514080564088, −4.782313845408766, −4.461236083981897, −3.391098151194264, −2.981838362034340, −2.225631154554469, −1.395567461025830, −0.5231253943039435,
0.5231253943039435, 1.395567461025830, 2.225631154554469, 2.981838362034340, 3.391098151194264, 4.461236083981897, 4.782313845408766, 5.405514080564088, 6.091494594150493, 6.742291263848008, 7.303171355703238, 7.736280590172995, 8.354205506209427, 9.038170942227119, 9.597756795260887, 9.965574263495164, 10.54601650233434, 11.09609320468096, 11.97129498345862, 12.13668846690863, 12.46386309720638, 13.31995557977405, 13.92325292679160, 14.22244408304544, 14.72265469674490
