| L(s) = 1 | − 2·4-s − 4·7-s + 2·13-s + 4·16-s + 8·28-s − 4·31-s − 37-s − 13·43-s + 9·49-s − 4·52-s − 61-s − 8·64-s + 11·67-s − 7·73-s − 13·79-s − 8·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s − 16·112-s + 113-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 0.554·13-s + 16-s + 1.51·28-s − 0.718·31-s − 0.164·37-s − 1.98·43-s + 9/7·49-s − 0.554·52-s − 0.128·61-s − 64-s + 1.34·67-s − 0.819·73-s − 1.46·79-s − 0.838·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s − 1.51·112-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5091158279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5091158279\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.73635185894381, −13.43040756504680, −13.09870591886181, −12.61653023986817, −12.16429371128532, −11.56263359996067, −10.90072110847116, −10.26469419365048, −9.993453576590887, −9.296116759684515, −9.199220876062275, −8.330456725370922, −8.176937564228890, −7.201132213864242, −6.810214713638526, −6.222760573530899, −5.638250950124409, −5.227185776442644, −4.409130904756836, −3.924489638578343, −3.295350453588827, −3.018120283758503, −1.963589756555912, −1.138436830856246, −0.2594549796857500,
0.2594549796857500, 1.138436830856246, 1.963589756555912, 3.018120283758503, 3.295350453588827, 3.924489638578343, 4.409130904756836, 5.227185776442644, 5.638250950124409, 6.222760573530899, 6.810214713638526, 7.201132213864242, 8.176937564228890, 8.330456725370922, 9.199220876062275, 9.296116759684515, 9.993453576590887, 10.26469419365048, 10.90072110847116, 11.56263359996067, 12.16429371128532, 12.61653023986817, 13.09870591886181, 13.43040756504680, 13.73635185894381
