| L(s) = 1 | + 3-s − 2·9-s + 6·11-s − 3·13-s + 23-s − 5·25-s − 5·27-s + 3·29-s + 7·31-s + 6·33-s − 8·37-s − 3·39-s + 11·41-s − 4·43-s − 47-s − 4·53-s + 12·59-s − 6·61-s − 12·67-s + 69-s − 5·71-s − 15·73-s − 5·75-s − 4·79-s + 81-s + 3·87-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.80·11-s − 0.832·13-s + 0.208·23-s − 25-s − 0.962·27-s + 0.557·29-s + 1.25·31-s + 1.04·33-s − 1.31·37-s − 0.480·39-s + 1.71·41-s − 0.609·43-s − 0.145·47-s − 0.549·53-s + 1.56·59-s − 0.768·61-s − 1.46·67-s + 0.120·69-s − 0.593·71-s − 1.75·73-s − 0.577·75-s − 0.450·79-s + 1/9·81-s + 0.321·87-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.38665211106055, −14.02264440817783, −13.45491138961983, −12.97699842582623, −12.10309927258049, −11.84622126443432, −11.65136584297337, −10.82925353588932, −10.24305513212671, −9.664355955193367, −9.280056060205222, −8.753845777293415, −8.387577871597528, −7.675904392848032, −7.212549314035068, −6.588836897779314, −6.056216612847292, −5.601035222453591, −4.656877154104016, −4.337699619676162, −3.570067731697421, −3.099875508440705, −2.408222458576652, −1.751353922649693, −1.004996329345890, 0,
1.004996329345890, 1.751353922649693, 2.408222458576652, 3.099875508440705, 3.570067731697421, 4.337699619676162, 4.656877154104016, 5.601035222453591, 6.056216612847292, 6.588836897779314, 7.212549314035068, 7.675904392848032, 8.387577871597528, 8.753845777293415, 9.280056060205222, 9.664355955193367, 10.24305513212671, 10.82925353588932, 11.65136584297337, 11.84622126443432, 12.10309927258049, 12.97699842582623, 13.45491138961983, 14.02264440817783, 14.38665211106055
