| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 9-s + 2·13-s − 4·15-s + 2·17-s − 4·19-s − 2·21-s − 25-s − 4·27-s + 2·29-s − 10·31-s + 2·35-s + 8·37-s + 4·39-s + 2·41-s − 4·43-s − 2·45-s − 6·47-s + 49-s + 4·51-s − 12·53-s − 8·57-s − 10·59-s − 6·61-s − 63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 0.917·19-s − 0.436·21-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.79·31-s + 0.338·35-s + 1.31·37-s + 0.640·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s + 0.560·51-s − 1.64·53-s − 1.05·57-s − 1.30·59-s − 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5841524931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5841524931\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.85058764447447, −12.64204657385624, −11.99511212138191, −11.46448058346217, −11.09375665674910, −10.63335382298224, −10.00070761290611, −9.539282707850460, −8.955537217019504, −8.827392687569129, −8.095485334334289, −7.719707343857871, −7.587832931200599, −6.779673855385644, −6.193528588745836, −5.879189076230113, −5.071439364793265, −4.421349361367768, −4.011921208306032, −3.446183751740447, −3.146301171416020, −2.566892500708475, −1.816060222823631, −1.333538967704276, −0.1866684238585408,
0.1866684238585408, 1.333538967704276, 1.816060222823631, 2.566892500708475, 3.146301171416020, 3.446183751740447, 4.011921208306032, 4.421349361367768, 5.071439364793265, 5.879189076230113, 6.193528588745836, 6.779673855385644, 7.587832931200599, 7.719707343857871, 8.095485334334289, 8.827392687569129, 8.955537217019504, 9.539282707850460, 10.00070761290611, 10.63335382298224, 11.09375665674910, 11.46448058346217, 11.99511212138191, 12.64204657385624, 12.85058764447447
