| L(s) = 1 | + 3·7-s + 11-s − 4·13-s + 3·17-s + 5·19-s − 4·23-s − 5·29-s − 7·31-s + 7·37-s + 8·41-s − 6·43-s − 8·47-s + 2·49-s + 9·53-s − 13·61-s − 12·67-s − 3·71-s + 6·73-s + 3·77-s − 4·83-s + 15·89-s − 12·91-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.301·11-s − 1.10·13-s + 0.727·17-s + 1.14·19-s − 0.834·23-s − 0.928·29-s − 1.25·31-s + 1.15·37-s + 1.24·41-s − 0.914·43-s − 1.16·47-s + 2/7·49-s + 1.23·53-s − 1.66·61-s − 1.46·67-s − 0.356·71-s + 0.702·73-s + 0.341·77-s − 0.439·83-s + 1.58·89-s − 1.25·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.79499915120432, −14.60895978517170, −14.23112143905342, −13.50708082864579, −13.04162182024108, −12.24295754274258, −11.98082091111900, −11.40660093055462, −11.01558624342566, −10.25681549042596, −9.756255598875734, −9.294431994091672, −8.708302805578531, −7.862968079984782, −7.578195224485980, −7.287863648655168, −6.235026861003365, −5.746437839939123, −5.071322378762405, −4.709020307338984, −3.910397806966780, −3.296393219151719, −2.448825332940621, −1.770294159673902, −1.124369454927755, 0,
1.124369454927755, 1.770294159673902, 2.448825332940621, 3.296393219151719, 3.910397806966780, 4.709020307338984, 5.071322378762405, 5.746437839939123, 6.235026861003365, 7.287863648655168, 7.578195224485980, 7.862968079984782, 8.708302805578531, 9.294431994091672, 9.756255598875734, 10.25681549042596, 11.01558624342566, 11.40660093055462, 11.98082091111900, 12.24295754274258, 13.04162182024108, 13.50708082864579, 14.23112143905342, 14.60895978517170, 14.79499915120432
