| L(s) = 1 | − 3·5-s + 4·7-s + 13-s − 19-s + 4·25-s − 6·29-s + 4·31-s − 12·35-s − 2·37-s − 43-s + 9·49-s − 3·53-s − 12·59-s − 2·61-s − 3·65-s − 4·67-s − 3·71-s + 14·73-s + 4·79-s − 9·83-s + 4·91-s + 3·95-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s + 0.277·13-s − 0.229·19-s + 4/5·25-s − 1.11·29-s + 0.718·31-s − 2.02·35-s − 0.328·37-s − 0.152·43-s + 9/7·49-s − 0.412·53-s − 1.56·59-s − 0.256·61-s − 0.372·65-s − 0.488·67-s − 0.356·71-s + 1.63·73-s + 0.450·79-s − 0.987·83-s + 0.419·91-s + 0.307·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.09291532501467, −12.57206058126129, −12.16002054751885, −11.65265260176918, −11.37015420844272, −10.97866358144780, −10.59396776370693, −9.987980861494934, −9.287999362803327, −8.816030142595752, −8.353728271105549, −7.944891753536859, −7.608767263402096, −7.223927693898036, −6.513796257272598, −5.987327733593480, −5.284816913838299, −4.866776044028416, −4.372241407007445, −3.967099411783974, −3.402629808520677, −2.796603739369488, −1.961781171986216, −1.524466238879549, −0.7625694198283359, 0,
0.7625694198283359, 1.524466238879549, 1.961781171986216, 2.796603739369488, 3.402629808520677, 3.967099411783974, 4.372241407007445, 4.866776044028416, 5.284816913838299, 5.987327733593480, 6.513796257272598, 7.223927693898036, 7.608767263402096, 7.944891753536859, 8.353728271105549, 8.816030142595752, 9.287999362803327, 9.987980861494934, 10.59396776370693, 10.97866358144780, 11.37015420844272, 11.65265260176918, 12.16002054751885, 12.57206058126129, 13.09291532501467
