| L(s) = 1 | + 5-s + 4·11-s + 4·17-s + 4·19-s + 6·23-s + 25-s − 6·29-s + 8·31-s + 2·37-s − 2·43-s + 6·53-s + 4·55-s − 4·59-s − 8·61-s + 2·67-s + 6·71-s + 2·73-s − 8·79-s − 4·83-s + 4·85-s − 12·89-s + 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.970·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.304·43-s + 0.824·53-s + 0.539·55-s − 0.520·59-s − 1.02·61-s + 0.244·67-s + 0.712·71-s + 0.234·73-s − 0.900·79-s − 0.439·83-s + 0.433·85-s − 1.27·89-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.487222689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.487222689\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.78079722439967, −12.16779071570928, −11.86368656529679, −11.43953875018147, −10.93265887622185, −10.44675376466746, −9.780012827626585, −9.606690675380999, −9.134910815232191, −8.649425045586683, −8.096688855436615, −7.574239095429922, −7.018583775108319, −6.735341194131344, −6.034385036671612, −5.687020957715140, −5.171552064113295, −4.581545951467363, −4.094607593647430, −3.349199388559130, −3.107512630218618, −2.395407605924173, −1.593558818583893, −1.183487126501197, −0.6238611576541841,
0.6238611576541841, 1.183487126501197, 1.593558818583893, 2.395407605924173, 3.107512630218618, 3.349199388559130, 4.094607593647430, 4.581545951467363, 5.171552064113295, 5.687020957715140, 6.034385036671612, 6.735341194131344, 7.018583775108319, 7.574239095429922, 8.096688855436615, 8.649425045586683, 9.134910815232191, 9.606690675380999, 9.780012827626585, 10.44675376466746, 10.93265887622185, 11.43953875018147, 11.86368656529679, 12.16779071570928, 12.78079722439967
