| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 6·11-s + 2·13-s + 16-s − 2·19-s − 20-s − 6·22-s + 4·23-s + 25-s + 2·26-s − 2·29-s + 8·31-s + 32-s − 8·37-s − 2·38-s − 40-s + 2·41-s − 43-s − 6·44-s + 4·46-s − 12·47-s − 7·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.80·11-s + 0.554·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 1.27·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.31·37-s − 0.324·38-s − 0.158·40-s + 0.312·41-s − 0.152·43-s − 0.904·44-s + 0.589·46-s − 1.75·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.095481635111723383242971288873, −7.35455468022359390412669744071, −6.57774585265733064811590754849, −5.78336562521772772343948042560, −4.96635905514274413574073817927, −4.47555495691855691996710902302, −3.28830456748531120744674893107, −2.82135329513580598823976803371, −1.60648775244345002228949754277, 0,
1.60648775244345002228949754277, 2.82135329513580598823976803371, 3.28830456748531120744674893107, 4.47555495691855691996710902302, 4.96635905514274413574073817927, 5.78336562521772772343948042560, 6.57774585265733064811590754849, 7.35455468022359390412669744071, 8.095481635111723383242971288873
