| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 4·7-s − 3·8-s + 9-s − 4·11-s − 12-s + 4·14-s − 16-s − 2·17-s + 18-s + 19-s + 4·21-s − 4·22-s + 4·23-s − 3·24-s + 27-s − 4·28-s − 2·29-s + 5·32-s − 4·33-s − 2·34-s − 36-s − 6·37-s + 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.872·21-s − 0.852·22-s + 0.834·23-s − 0.612·24-s + 0.192·27-s − 0.755·28-s − 0.371·29-s + 0.883·32-s − 0.696·33-s − 0.342·34-s − 1/6·36-s − 0.986·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.23089295422820, −12.87276938709437, −12.25027223036791, −11.82051838912817, −11.29339806561966, −10.91652266982487, −10.33142489682431, −9.910014359624367, −9.280014850012257, −8.781773766281466, −8.439532749339781, −8.003655604149578, −7.613332580982869, −6.948379838790289, −6.506816433613589, −5.514319397541193, −5.344940074828892, −4.971111232953020, −4.379534492013383, −4.002698679270767, −3.264369546532385, −2.830654581286378, −2.194351699371337, −1.648769507929204, −0.8585644090574967, 0,
0.8585644090574967, 1.648769507929204, 2.194351699371337, 2.830654581286378, 3.264369546532385, 4.002698679270767, 4.379534492013383, 4.971111232953020, 5.344940074828892, 5.514319397541193, 6.506816433613589, 6.948379838790289, 7.613332580982869, 8.003655604149578, 8.439532749339781, 8.781773766281466, 9.280014850012257, 9.910014359624367, 10.33142489682431, 10.91652266982487, 11.29339806561966, 11.82051838912817, 12.25027223036791, 12.87276938709437, 13.23089295422820
