| L(s) = 1 | − 2·4-s − 5-s + 13-s + 4·16-s + 3·17-s + 8·19-s + 2·20-s + 3·23-s + 25-s − 6·29-s − 4·31-s + 2·37-s − 9·41-s − 4·43-s − 6·47-s − 2·52-s − 9·53-s + 15·59-s + 5·61-s − 8·64-s − 65-s − 13·67-s − 6·68-s + 15·71-s − 13·73-s − 16·76-s + 8·79-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.277·13-s + 16-s + 0.727·17-s + 1.83·19-s + 0.447·20-s + 0.625·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 1.40·41-s − 0.609·43-s − 0.875·47-s − 0.277·52-s − 1.23·53-s + 1.95·59-s + 0.640·61-s − 64-s − 0.124·65-s − 1.58·67-s − 0.727·68-s + 1.78·71-s − 1.52·73-s − 1.83·76-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−15.40260234663807, −14.74554475129629, −14.49767530714130, −13.84596000773737, −13.26062100611423, −12.97560118380935, −12.30081323698870, −11.57274741091428, −11.44571738516112, −10.48671075944861, −9.977962734507026, −9.461045475867676, −8.993863687238278, −8.355300394585740, −7.764456660438483, −7.379533635044220, −6.622559079188114, −5.763446602841568, −5.154060940580678, −4.933242662178190, −3.834323328071180, −3.535926129583251, −2.928349315939761, −1.639169043504996, −0.9667387989568812, 0,
0.9667387989568812, 1.639169043504996, 2.928349315939761, 3.535926129583251, 3.834323328071180, 4.933242662178190, 5.154060940580678, 5.763446602841568, 6.622559079188114, 7.379533635044220, 7.764456660438483, 8.355300394585740, 8.993863687238278, 9.461045475867676, 9.977962734507026, 10.48671075944861, 11.44571738516112, 11.57274741091428, 12.30081323698870, 12.97560118380935, 13.26062100611423, 13.84596000773737, 14.49767530714130, 14.74554475129629, 15.40260234663807
