| L(s) = 1 | − 4·5-s − 7-s − 2·11-s + 4·19-s + 6·23-s + 11·25-s + 10·29-s + 8·31-s + 4·35-s − 10·37-s − 4·41-s − 8·43-s − 4·47-s + 49-s − 10·53-s + 8·55-s + 8·59-s − 6·61-s − 4·67-s + 14·71-s − 6·73-s + 2·77-s + 4·79-s − 12·83-s + 4·89-s − 16·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s − 0.603·11-s + 0.917·19-s + 1.25·23-s + 11/5·25-s + 1.85·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s − 0.624·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s − 1.37·53-s + 1.07·55-s + 1.04·59-s − 0.768·61-s − 0.488·67-s + 1.66·71-s − 0.702·73-s + 0.227·77-s + 0.450·79-s − 1.31·83-s + 0.423·89-s − 1.64·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9972440649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9972440649\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−12.38596196223533, −12.07644563971250, −11.77132227143348, −11.28067370706876, −10.83296963217597, −10.31140478050295, −9.981117365297801, −9.376778410928358, −8.611568764411699, −8.487505471110253, −8.017112929904898, −7.534414019441405, −7.002757869646481, −6.698600224680640, −6.233559356752995, −5.245639641209005, −4.932453878124500, −4.662399817152800, −3.895205913188079, −3.395099572782120, −2.992451000659705, −2.687492349564823, −1.551318242082204, −0.9455702942158740, −0.3214605275735955,
0.3214605275735955, 0.9455702942158740, 1.551318242082204, 2.687492349564823, 2.992451000659705, 3.395099572782120, 3.895205913188079, 4.662399817152800, 4.932453878124500, 5.245639641209005, 6.233559356752995, 6.698600224680640, 7.002757869646481, 7.534414019441405, 8.017112929904898, 8.487505471110253, 8.611568764411699, 9.376778410928358, 9.981117365297801, 10.31140478050295, 10.83296963217597, 11.28067370706876, 11.77132227143348, 12.07644563971250, 12.38596196223533
