| L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·11-s − 6·13-s + 2·15-s + 2·17-s − 4·19-s + 23-s − 25-s + 27-s − 6·29-s + 4·33-s + 2·37-s − 6·39-s + 41-s + 4·43-s + 2·45-s + 8·47-s − 7·49-s + 2·51-s − 6·53-s + 8·55-s − 4·57-s − 4·59-s − 6·61-s − 12·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.328·37-s − 0.960·39-s + 0.156·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s + 0.280·51-s − 0.824·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s − 0.768·61-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.47170928542467, −12.84681096467995, −12.47374063207069, −12.20822062089045, −11.50494598815296, −10.98004321906440, −10.52492506167761, −9.806124753600741, −9.544513454166155, −9.318868423599595, −8.793840411727727, −7.951314014701819, −7.792413840272646, −7.065353605531931, −6.628551434850052, −6.181333141568654, −5.544730918540167, −5.053108010538427, −4.463064506292812, −3.894815105599595, −3.409883642299504, −2.559247067765973, −2.221820959099613, −1.692021242835929, −0.9673896113520894, 0,
0.9673896113520894, 1.692021242835929, 2.221820959099613, 2.559247067765973, 3.409883642299504, 3.894815105599595, 4.463064506292812, 5.053108010538427, 5.544730918540167, 6.181333141568654, 6.628551434850052, 7.065353605531931, 7.792413840272646, 7.951314014701819, 8.793840411727727, 9.318868423599595, 9.544513454166155, 9.806124753600741, 10.52492506167761, 10.98004321906440, 11.50494598815296, 12.20822062089045, 12.47374063207069, 12.84681096467995, 13.47170928542467
