| L(s) = 1 | + 2·2-s + 2·4-s + 7-s + 3·13-s + 2·14-s − 4·16-s − 2·17-s + 4·19-s + 4·23-s + 6·26-s + 2·28-s + 2·29-s + 31-s − 8·32-s − 4·34-s + 7·37-s + 8·38-s + 6·41-s + 7·43-s + 8·46-s − 8·47-s + 49-s + 6·52-s − 4·53-s + 4·58-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s + 0.832·13-s + 0.534·14-s − 16-s − 0.485·17-s + 0.917·19-s + 0.834·23-s + 1.17·26-s + 0.377·28-s + 0.371·29-s + 0.179·31-s − 1.41·32-s − 0.685·34-s + 1.15·37-s + 1.29·38-s + 0.937·41-s + 1.06·43-s + 1.17·46-s − 1.16·47-s + 1/7·49-s + 0.832·52-s − 0.549·53-s + 0.525·58-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.578908461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.578908461\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−8.155623586579538706757570672036, −7.43176687731826870361366039115, −6.50214677704479677044463853057, −6.05230827210832654104884385745, −5.16883676639962667180243946464, −4.65444786211788835134520228819, −3.84661147971485487844529928407, −3.12038194314791806442671455008, −2.28238812168272882685413248311, −0.995779494632046062759647561951,
0.995779494632046062759647561951, 2.28238812168272882685413248311, 3.12038194314791806442671455008, 3.84661147971485487844529928407, 4.65444786211788835134520228819, 5.16883676639962667180243946464, 6.05230827210832654104884385745, 6.50214677704479677044463853057, 7.43176687731826870361366039115, 8.155623586579538706757570672036
