| L(s) = 1 | − 2·4-s + 4·16-s − 9·23-s − 5·31-s − 7·37-s − 12·47-s − 7·49-s + 6·53-s + 15·59-s − 8·64-s − 13·67-s + 3·71-s + 9·89-s + 18·92-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 1.87·23-s − 0.898·31-s − 1.15·37-s − 1.75·47-s − 49-s + 0.824·53-s + 1.95·59-s − 64-s − 1.58·67-s + 0.356·71-s + 0.953·89-s + 1.87·92-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7477359857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7477359857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−15.16863713523087, −14.50930387685395, −14.31404645708268, −13.60077379396845, −13.21887796171982, −12.65960018173310, −12.07163879590727, −11.63419297861527, −10.87271808682258, −10.19818999519871, −9.863152858559874, −9.306577194639919, −8.565187343230728, −8.277777762682828, −7.617528098615488, −6.949153902280101, −6.183610589076477, −5.613700607862709, −5.042841785202149, −4.382740074763934, −3.744852216726785, −3.270994249527195, −2.170032500137220, −1.483700781994962, −0.3441940286052214,
0.3441940286052214, 1.483700781994962, 2.170032500137220, 3.270994249527195, 3.744852216726785, 4.382740074763934, 5.042841785202149, 5.613700607862709, 6.183610589076477, 6.949153902280101, 7.617528098615488, 8.277777762682828, 8.565187343230728, 9.306577194639919, 9.863152858559874, 10.19818999519871, 10.87271808682258, 11.63419297861527, 12.07163879590727, 12.65960018173310, 13.21887796171982, 13.60077379396845, 14.31404645708268, 14.50930387685395, 15.16863713523087
