| L(s) = 1 | − 3-s + 5-s + 9-s − 3·11-s + 13-s − 15-s + 19-s + 2·23-s + 25-s − 27-s + 9·29-s + 3·33-s − 10·37-s − 39-s − 41-s + 45-s + 4·47-s − 7·49-s + 4·53-s − 3·55-s − 57-s + 5·59-s + 7·61-s + 65-s + 4·67-s − 2·69-s + 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.258·15-s + 0.229·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 0.522·33-s − 1.64·37-s − 0.160·39-s − 0.156·41-s + 0.149·45-s + 0.583·47-s − 49-s + 0.549·53-s − 0.404·55-s − 0.132·57-s + 0.650·59-s + 0.896·61-s + 0.124·65-s + 0.488·67-s − 0.240·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.296111219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.296111219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.76415109676262, −12.68819233385895, −11.96197106577406, −11.53951642777808, −11.15093403994540, −10.42090451868503, −10.25323307170836, −9.956086523613413, −9.136499057334038, −8.721319648964037, −8.313048752327702, −7.663111805950718, −7.219425164279975, −6.562419843637813, −6.394036523391040, −5.521159623731716, −5.346755639886506, −4.789872070048637, −4.299758351587660, −3.462832440412220, −3.086327381350779, −2.326880225768235, −1.851619372244018, −1.011493863530628, −0.5015894617656842,
0.5015894617656842, 1.011493863530628, 1.851619372244018, 2.326880225768235, 3.086327381350779, 3.462832440412220, 4.299758351587660, 4.789872070048637, 5.346755639886506, 5.521159623731716, 6.394036523391040, 6.562419843637813, 7.219425164279975, 7.663111805950718, 8.313048752327702, 8.721319648964037, 9.136499057334038, 9.956086523613413, 10.25323307170836, 10.42090451868503, 11.15093403994540, 11.53951642777808, 11.96197106577406, 12.68819233385895, 12.76415109676262
