| L(s) = 1 | + 2·7-s − 4·11-s + 13-s − 8·17-s + 2·19-s + 4·23-s − 8·29-s − 10·31-s − 6·37-s + 6·41-s − 8·43-s − 8·47-s − 3·49-s + 12·53-s + 4·59-s + 10·61-s + 2·67-s − 6·73-s − 8·77-s − 12·79-s − 4·83-s − 6·89-s + 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s + 0.277·13-s − 1.94·17-s + 0.458·19-s + 0.834·23-s − 1.48·29-s − 1.79·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 1.16·47-s − 3/7·49-s + 1.64·53-s + 0.520·59-s + 1.28·61-s + 0.244·67-s − 0.702·73-s − 0.911·77-s − 1.35·79-s − 0.439·83-s − 0.635·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061746018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061746018\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.73965818489829, −14.14097515618142, −13.28430312427330, −13.11622789953170, −12.88712669435242, −11.83531169347452, −11.41672290423850, −10.92324403764814, −10.71460499021361, −9.895415125024798, −9.316658793049544, −8.704251145256600, −8.389374118682691, −7.693206058198613, −7.038971744613669, −6.857115549817335, −5.741989065247096, −5.388336304271782, −4.885364089617596, −4.189538271691552, −3.559474524289517, −2.773315007187946, −2.049006932040848, −1.592307928121322, −0.3473046434064016,
0.3473046434064016, 1.592307928121322, 2.049006932040848, 2.773315007187946, 3.559474524289517, 4.189538271691552, 4.885364089617596, 5.388336304271782, 5.741989065247096, 6.857115549817335, 7.038971744613669, 7.693206058198613, 8.389374118682691, 8.704251145256600, 9.316658793049544, 9.895415125024798, 10.71460499021361, 10.92324403764814, 11.41672290423850, 11.83531169347452, 12.88712669435242, 13.11622789953170, 13.28430312427330, 14.14097515618142, 14.73965818489829
