| L(s) = 1 | + 2-s − 4-s − 2·5-s + 4·7-s − 3·8-s − 2·10-s + 6·13-s + 4·14-s − 16-s − 2·17-s + 4·19-s + 2·20-s − 4·23-s − 25-s + 6·26-s − 4·28-s + 2·29-s − 8·31-s + 5·32-s − 2·34-s − 8·35-s + 37-s + 4·38-s + 6·40-s + 6·41-s + 4·43-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.51·7-s − 1.06·8-s − 0.632·10-s + 1.66·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.447·20-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.755·28-s + 0.371·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s − 1.35·35-s + 0.164·37-s + 0.648·38-s + 0.948·40-s + 0.937·41-s + 0.609·43-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40293 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40293 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.692914843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.692914843\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 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.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−14.70281656904550, −14.14480383532033, −13.91970403315205, −13.20442742599889, −12.87221355737574, −12.05460030218104, −11.63639894881842, −11.34946872922500, −10.81017684068014, −10.16697188149306, −9.242685293088166, −8.881108689652097, −8.320990286178735, −7.848076778637128, −7.445731628371335, −6.504589981223897, −5.798803993968421, −5.486732202386083, −4.578846929733445, −4.375797609820970, −3.655548971227552, −3.299217225230795, −2.178534871488251, −1.397988114661482, −0.5723659022503378,
0.5723659022503378, 1.397988114661482, 2.178534871488251, 3.299217225230795, 3.655548971227552, 4.375797609820970, 4.578846929733445, 5.486732202386083, 5.798803993968421, 6.504589981223897, 7.445731628371335, 7.848076778637128, 8.320990286178735, 8.881108689652097, 9.242685293088166, 10.16697188149306, 10.81017684068014, 11.34946872922500, 11.63639894881842, 12.05460030218104, 12.87221355737574, 13.20442742599889, 13.91970403315205, 14.14480383532033, 14.70281656904550
