| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s + 4·11-s − 12-s + 2·13-s + 4·14-s + 16-s − 18-s − 4·19-s + 4·21-s − 4·22-s − 4·23-s + 24-s − 2·26-s − 27-s − 4·28-s − 2·29-s − 4·31-s − 32-s − 4·33-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.94624032407187, −14.61788915276389, −13.85217284918695, −13.26355712238266, −12.77977196809899, −12.29842635954041, −11.80925965520651, −11.29999306224882, −10.59624935142215, −10.31210762823745, −9.641887911668948, −9.190714325982043, −8.818071254562338, −8.117507026897114, −7.313663204143508, −6.860140977238800, −6.275971740603037, −6.088437144965171, −5.364088925378243, −4.344050281517369, −3.745819758990863, −3.365422427924331, −2.333855853766632, −1.668661509520086, −0.7569552069784493, 0,
0.7569552069784493, 1.668661509520086, 2.333855853766632, 3.365422427924331, 3.745819758990863, 4.344050281517369, 5.364088925378243, 6.088437144965171, 6.275971740603037, 6.860140977238800, 7.313663204143508, 8.117507026897114, 8.818071254562338, 9.190714325982043, 9.641887911668948, 10.31210762823745, 10.59624935142215, 11.29999306224882, 11.80925965520651, 12.29842635954041, 12.77977196809899, 13.26355712238266, 13.85217284918695, 14.61788915276389, 14.94624032407187
