| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s − 3·11-s + 2·12-s + 13-s + 14-s + 16-s + 18-s − 4·19-s + 2·21-s − 3·22-s − 3·23-s + 2·24-s + 26-s − 4·27-s + 28-s − 6·29-s − 10·31-s + 32-s − 6·33-s + 36-s + 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.436·21-s − 0.639·22-s − 0.625·23-s + 0.408·24-s + 0.196·26-s − 0.769·27-s + 0.188·28-s − 1.11·29-s − 1.79·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.27084792184944, −15.65843771626078, −14.95057213748267, −14.64958041438992, −14.41415805115677, −13.32955195002831, −13.27895942462950, −12.82739991956076, −11.89543756689208, −11.31635216675895, −10.83223552812833, −10.10248652774804, −9.509395413673397, −8.703860626971002, −8.331014292716160, −7.626085865939863, −7.268235067607759, −6.253548896838430, −5.670255220092576, −5.018806912167490, −4.158497457427303, −3.655365000550460, −2.911499550562403, −2.196182017799202, −1.670330902574769, 0,
1.670330902574769, 2.196182017799202, 2.911499550562403, 3.655365000550460, 4.158497457427303, 5.018806912167490, 5.670255220092576, 6.253548896838430, 7.268235067607759, 7.626085865939863, 8.331014292716160, 8.703860626971002, 9.509395413673397, 10.10248652774804, 10.83223552812833, 11.31635216675895, 11.89543756689208, 12.82739991956076, 13.27895942462950, 13.32955195002831, 14.41415805115677, 14.64958041438992, 14.95057213748267, 15.65843771626078, 16.27084792184944
