| L(s) = 1 | − 7-s − 5·11-s + 6·13-s + 7·17-s − 6·19-s − 2·23-s − 8·31-s + 8·37-s − 8·41-s − 43-s + 8·47-s − 6·49-s − 3·53-s − 4·59-s + 11·61-s + 67-s + 15·71-s − 4·73-s + 5·77-s + 16·83-s − 2·89-s − 6·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s + 1.66·13-s + 1.69·17-s − 1.37·19-s − 0.417·23-s − 1.43·31-s + 1.31·37-s − 1.24·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s − 0.412·53-s − 0.520·59-s + 1.40·61-s + 0.122·67-s + 1.78·71-s − 0.468·73-s + 0.569·77-s + 1.75·83-s − 0.211·89-s − 0.628·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−15.81681127675234, −15.13385421922996, −14.66843203306105, −14.03130545171459, −13.40030214341074, −12.96816910783985, −12.64080204070032, −11.94510995022998, −11.15560240941051, −10.77840002395147, −10.27629674727252, −9.747036780128620, −9.025027967887600, −8.346437472568273, −7.975664575303025, −7.442808747606050, −6.508101431911560, −6.084220172181315, −5.457205032150204, −4.940647334560857, −3.834651789357238, −3.586937168128345, −2.713898753716992, −1.958033529607061, −1.022955216901510, 0,
1.022955216901510, 1.958033529607061, 2.713898753716992, 3.586937168128345, 3.834651789357238, 4.940647334560857, 5.457205032150204, 6.084220172181315, 6.508101431911560, 7.442808747606050, 7.975664575303025, 8.346437472568273, 9.025027967887600, 9.747036780128620, 10.27629674727252, 10.77840002395147, 11.15560240941051, 11.94510995022998, 12.64080204070032, 12.96816910783985, 13.40030214341074, 14.03130545171459, 14.66843203306105, 15.13385421922996, 15.81681127675234
