| L(s) = 1 | − 5-s + 3·7-s + 11-s + 13-s + 17-s + 5·19-s − 4·23-s + 25-s − 7·29-s − 5·31-s − 3·35-s − 3·37-s + 8·41-s + 6·43-s − 4·47-s + 2·49-s + 9·53-s − 55-s − 10·59-s − 11·61-s − 65-s − 4·67-s − 9·71-s − 2·73-s + 3·77-s − 4·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.301·11-s + 0.277·13-s + 0.242·17-s + 1.14·19-s − 0.834·23-s + 1/5·25-s − 1.29·29-s − 0.898·31-s − 0.507·35-s − 0.493·37-s + 1.24·41-s + 0.914·43-s − 0.583·47-s + 2/7·49-s + 1.23·53-s − 0.134·55-s − 1.30·59-s − 1.40·61-s − 0.124·65-s − 0.488·67-s − 1.06·71-s − 0.234·73-s + 0.341·77-s − 0.450·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.95699546426870, −13.68647735572252, −12.87449502842268, −12.50300672444302, −11.89768763364184, −11.50980998026890, −11.13498627728956, −10.69191786948402, −10.02754735657342, −9.518868123983434, −8.836183374153568, −8.670160052199475, −7.753949051184118, −7.491719565940414, −7.305525355290776, −6.237704709961209, −5.794974394896515, −5.356632435757568, −4.588526895768475, −4.262070514258581, −3.535815001885440, −3.092699593564949, −2.127006443407910, −1.612132072623374, −0.9756852331455902, 0,
0.9756852331455902, 1.612132072623374, 2.127006443407910, 3.092699593564949, 3.535815001885440, 4.262070514258581, 4.588526895768475, 5.356632435757568, 5.794974394896515, 6.237704709961209, 7.305525355290776, 7.491719565940414, 7.753949051184118, 8.670160052199475, 8.836183374153568, 9.518868123983434, 10.02754735657342, 10.69191786948402, 11.13498627728956, 11.50980998026890, 11.89768763364184, 12.50300672444302, 12.87449502842268, 13.68647735572252, 13.95699546426870
