| L(s) = 1 | − 7-s + 6·11-s − 4·17-s − 3·23-s − 29-s + 4·31-s + 10·37-s − 3·41-s + 4·43-s − 7·47-s − 6·49-s − 6·53-s − 61-s + 7·67-s + 2·71-s − 10·73-s − 6·77-s + 6·79-s − 17·83-s + 9·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.970·17-s − 0.625·23-s − 0.185·29-s + 0.718·31-s + 1.64·37-s − 0.468·41-s + 0.609·43-s − 1.02·47-s − 6/7·49-s − 0.824·53-s − 0.128·61-s + 0.855·67-s + 0.237·71-s − 1.17·73-s − 0.683·77-s + 0.675·79-s − 1.86·83-s + 0.953·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.05097305387290, −12.63627760639376, −11.96585552499005, −11.77887924123070, −11.09339584572292, −11.04886503777798, −10.09798539048589, −9.754177433962690, −9.395791642709859, −8.928278331446010, −8.425235562070492, −7.972523952696829, −7.352860345550290, −6.768032454369396, −6.395266508699811, −6.144370023731738, −5.503375965141988, −4.588978968792641, −4.465003703337993, −3.830587048893800, −3.332371747931861, −2.712251334148282, −2.021837706187276, −1.474013617494921, −0.8217267471352672, 0,
0.8217267471352672, 1.474013617494921, 2.021837706187276, 2.712251334148282, 3.332371747931861, 3.830587048893800, 4.465003703337993, 4.588978968792641, 5.503375965141988, 6.144370023731738, 6.395266508699811, 6.768032454369396, 7.352860345550290, 7.972523952696829, 8.425235562070492, 8.928278331446010, 9.395791642709859, 9.754177433962690, 10.09798539048589, 11.04886503777798, 11.09339584572292, 11.77887924123070, 11.96585552499005, 12.63627760639376, 13.05097305387290
