| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s + 2·13-s − 2·15-s + 6·17-s − 8·19-s − 4·21-s − 23-s − 25-s − 27-s − 6·29-s − 8·31-s + 8·35-s + 2·37-s − 2·39-s + 41-s + 12·43-s + 2·45-s + 8·47-s + 9·49-s − 6·51-s − 10·53-s + 8·57-s − 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s − 0.872·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.35·35-s + 0.328·37-s − 0.320·39-s + 0.156·41-s + 1.82·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s + 1.05·57-s − 0.520·59-s − 0.768·61-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 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.33915983742662, −12.78138283628728, −12.56089619424433, −11.99636585382305, −11.25979812678187, −11.13245846228260, −10.59477744618114, −10.29427149235007, −9.584027368371303, −9.106836361309066, −8.715276585921366, −7.999207885792532, −7.635706899250210, −7.272908485474391, −6.387903331969816, −5.847575449191557, −5.759695018682027, −5.157378581447292, −4.510202966726244, −4.087663926574039, −3.515894228461283, −2.552735320018563, −1.943449048948982, −1.605546670943707, −0.9893559673784060, 0,
0.9893559673784060, 1.605546670943707, 1.943449048948982, 2.552735320018563, 3.515894228461283, 4.087663926574039, 4.510202966726244, 5.157378581447292, 5.759695018682027, 5.847575449191557, 6.387903331969816, 7.272908485474391, 7.635706899250210, 7.999207885792532, 8.715276585921366, 9.106836361309066, 9.584027368371303, 10.29427149235007, 10.59477744618114, 11.13245846228260, 11.25979812678187, 11.99636585382305, 12.56089619424433, 12.78138283628728, 13.33915983742662
