| L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s + 5·13-s − 3·15-s + 17-s − 6·19-s − 21-s + 4·23-s + 4·25-s + 27-s + 29-s + 6·31-s + 3·35-s − 3·37-s + 5·39-s + 9·41-s + 10·43-s − 3·45-s + 10·47-s + 49-s + 51-s + 3·53-s − 6·57-s + 2·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.38·13-s − 0.774·15-s + 0.242·17-s − 1.37·19-s − 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s + 0.185·29-s + 1.07·31-s + 0.507·35-s − 0.493·37-s + 0.800·39-s + 1.40·41-s + 1.52·43-s − 0.447·45-s + 1.45·47-s + 1/7·49-s + 0.140·51-s + 0.412·53-s − 0.794·57-s + 0.260·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.273580472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273580472\) |
| \(L(\frac{3}{2})\) |
|
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 | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−14.75812486899833, −14.35465412174505, −13.69748554917827, −13.22903638515481, −12.50566567065365, −12.44952505649066, −11.52210749592867, −11.15944599549315, −10.59716796894244, −10.17737053756980, −9.188181971257427, −8.911198475610054, −8.302479766827623, −7.981442486081686, −7.243728612470057, −6.822743185984423, −6.107036981216339, −5.560323539798254, −4.468176008765724, −4.131470814782091, −3.702509416519987, −2.928265325412799, −2.405707611435908, −1.243070098831822, −0.5910429549601690,
0.5910429549601690, 1.243070098831822, 2.405707611435908, 2.928265325412799, 3.702509416519987, 4.131470814782091, 4.468176008765724, 5.560323539798254, 6.107036981216339, 6.822743185984423, 7.243728612470057, 7.981442486081686, 8.302479766827623, 8.911198475610054, 9.188181971257427, 10.17737053756980, 10.59716796894244, 11.15944599549315, 11.52210749592867, 12.44952505649066, 12.50566567065365, 13.22903638515481, 13.69748554917827, 14.35465412174505, 14.75812486899833
