| L(s) = 1 | + 3·5-s + 7-s − 3·11-s + 5·13-s − 2·19-s + 4·25-s + 2·31-s + 3·35-s + 37-s − 5·43-s − 6·47-s + 49-s + 3·53-s − 9·55-s − 12·59-s − 8·61-s + 15·65-s − 11·67-s + 12·71-s + 7·73-s − 3·77-s − 7·79-s − 3·83-s + 9·89-s + 5·91-s − 6·95-s − 17·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.904·11-s + 1.38·13-s − 0.458·19-s + 4/5·25-s + 0.359·31-s + 0.507·35-s + 0.164·37-s − 0.762·43-s − 0.875·47-s + 1/7·49-s + 0.412·53-s − 1.21·55-s − 1.56·59-s − 1.02·61-s + 1.86·65-s − 1.34·67-s + 1.42·71-s + 0.819·73-s − 0.341·77-s − 0.787·79-s − 0.329·83-s + 0.953·89-s + 0.524·91-s − 0.615·95-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.06634273229327, −12.60416600185390, −12.06661937902711, −11.50251685547978, −10.83142455534285, −10.79435682062383, −10.22267704735532, −9.738289000873431, −9.316555541547831, −8.757593946394443, −8.381927273591264, −7.894425985264424, −7.427969550944863, −6.590777082113366, −6.334359774541399, −5.943009534255116, −5.267395879856327, −5.091012789126603, −4.320755966699843, −3.837402742746184, −2.968520959261147, −2.772852254133635, −1.785670595502071, −1.714344733663583, −0.9229091988630348, 0,
0.9229091988630348, 1.714344733663583, 1.785670595502071, 2.772852254133635, 2.968520959261147, 3.837402742746184, 4.320755966699843, 5.091012789126603, 5.267395879856327, 5.943009534255116, 6.334359774541399, 6.590777082113366, 7.427969550944863, 7.894425985264424, 8.381927273591264, 8.757593946394443, 9.316555541547831, 9.738289000873431, 10.22267704735532, 10.79435682062383, 10.83142455534285, 11.50251685547978, 12.06661937902711, 12.60416600185390, 13.06634273229327
