| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s + 3·11-s − 4·16-s − 5·17-s + 2·19-s + 2·20-s + 6·22-s − 4·23-s + 25-s − 7·29-s − 10·31-s − 8·32-s − 10·34-s − 2·37-s + 4·38-s − 10·43-s + 6·44-s − 8·46-s + 3·47-s + 2·50-s − 4·53-s + 3·55-s − 14·58-s − 4·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 0.904·11-s − 16-s − 1.21·17-s + 0.458·19-s + 0.447·20-s + 1.27·22-s − 0.834·23-s + 1/5·25-s − 1.29·29-s − 1.79·31-s − 1.41·32-s − 1.71·34-s − 0.328·37-s + 0.648·38-s − 1.52·43-s + 0.904·44-s − 1.17·46-s + 0.437·47-s + 0.282·50-s − 0.549·53-s + 0.404·55-s − 1.83·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.216550471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.216550471\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.60695988075955, −12.20587965570086, −11.50288471772964, −11.44459745024717, −10.92686999169159, −10.34721867990445, −9.679182761793781, −9.367492527965288, −8.901339325993405, −8.517819495317897, −7.784567615550922, −7.079547678363228, −6.924056820539675, −6.317518991779951, −5.871439409464341, −5.486535951309407, −5.003859914752559, −4.461145127197476, −3.911502103994186, −3.652352207732931, −3.067496994458736, −2.374075731583219, −1.848517750909112, −1.481618899220255, −0.2671523593598539,
0.2671523593598539, 1.481618899220255, 1.848517750909112, 2.374075731583219, 3.067496994458736, 3.652352207732931, 3.911502103994186, 4.461145127197476, 5.003859914752559, 5.486535951309407, 5.871439409464341, 6.317518991779951, 6.924056820539675, 7.079547678363228, 7.784567615550922, 8.517819495317897, 8.901339325993405, 9.367492527965288, 9.679182761793781, 10.34721867990445, 10.92686999169159, 11.44459745024717, 11.50288471772964, 12.20587965570086, 12.60695988075955
