| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 5·11-s + 12-s + 3·13-s + 16-s − 3·17-s + 2·18-s − 6·19-s + 5·22-s + 4·23-s − 24-s − 3·26-s − 5·27-s − 6·29-s − 7·31-s − 32-s − 5·33-s + 3·34-s − 2·36-s − 3·37-s + 6·38-s + 3·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 1.50·11-s + 0.288·12-s + 0.832·13-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.37·19-s + 1.06·22-s + 0.834·23-s − 0.204·24-s − 0.588·26-s − 0.962·27-s − 1.11·29-s − 1.25·31-s − 0.176·32-s − 0.870·33-s + 0.514·34-s − 1/3·36-s − 0.493·37-s + 0.973·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−13.31011835162462, −12.88293347253813, −12.58583089923517, −11.74206188228670, −11.16951355874881, −11.01518896919868, −10.49989045234846, −10.13401300078954, −9.292269133161789, −9.009744253394580, −8.632210733781410, −8.154723606120301, −7.776786578451167, −7.207757013919099, −6.694483374340663, −6.096258757526524, −5.587077005401190, −5.136492534085704, −4.442935677426163, −3.669189388760274, −3.282836654182971, −2.647420573036764, −2.046065944038832, −1.772833173494605, −0.5792667889867707, 0,
0.5792667889867707, 1.772833173494605, 2.046065944038832, 2.647420573036764, 3.282836654182971, 3.669189388760274, 4.442935677426163, 5.136492534085704, 5.587077005401190, 6.096258757526524, 6.694483374340663, 7.207757013919099, 7.776786578451167, 8.154723606120301, 8.632210733781410, 9.009744253394580, 9.292269133161789, 10.13401300078954, 10.49989045234846, 11.01518896919868, 11.16951355874881, 11.74206188228670, 12.58583089923517, 12.88293347253813, 13.31011835162462
