| L(s) = 1 | − 3·5-s − 6·11-s + 13-s + 3·17-s + 2·19-s − 6·23-s + 4·25-s + 9·29-s − 10·31-s − 7·37-s + 6·41-s + 4·43-s − 6·47-s + 6·53-s + 18·55-s + 6·59-s − 11·61-s − 3·65-s − 2·67-s + 12·71-s + 7·73-s − 2·79-s − 6·83-s − 9·85-s + 3·89-s − 6·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.458·19-s − 1.25·23-s + 4/5·25-s + 1.67·29-s − 1.79·31-s − 1.15·37-s + 0.937·41-s + 0.609·43-s − 0.875·47-s + 0.824·53-s + 2.42·55-s + 0.781·59-s − 1.40·61-s − 0.372·65-s − 0.244·67-s + 1.42·71-s + 0.819·73-s − 0.225·79-s − 0.658·83-s − 0.976·85-s + 0.317·89-s − 0.615·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.56462404788048, −13.86160907540794, −13.66331195230807, −12.79902977151397, −12.37058951876398, −12.17102694175399, −11.44806895331762, −10.88173955576624, −10.62030252463516, −9.955438177908243, −9.500230981300184, −8.552177149195855, −8.272099281116087, −7.784800364450164, −7.402970858459258, −6.864668219619294, −5.986544564210616, −5.433963050227389, −5.011102159539594, −4.232641820064368, −3.750504334942809, −3.123952090635235, −2.578815144350263, −1.715277664472532, −0.6828489772743290, 0,
0.6828489772743290, 1.715277664472532, 2.578815144350263, 3.123952090635235, 3.750504334942809, 4.232641820064368, 5.011102159539594, 5.433963050227389, 5.986544564210616, 6.864668219619294, 7.402970858459258, 7.784800364450164, 8.272099281116087, 8.552177149195855, 9.500230981300184, 9.955438177908243, 10.62030252463516, 10.88173955576624, 11.44806895331762, 12.17102694175399, 12.37058951876398, 12.79902977151397, 13.66331195230807, 13.86160907540794, 14.56462404788048
