| L(s) = 1 | + 6·11-s − 2·13-s − 6·17-s + 2·19-s + 23-s − 5·25-s + 6·29-s + 8·31-s + 8·37-s + 6·41-s − 2·43-s + 12·53-s − 8·61-s + 10·67-s − 14·73-s − 8·79-s − 6·83-s + 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.80·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.208·23-s − 25-s + 1.11·29-s + 1.43·31-s + 1.31·37-s + 0.937·41-s − 0.304·43-s + 1.64·53-s − 1.02·61-s + 1.22·67-s − 1.63·73-s − 0.900·79-s − 0.658·83-s + 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.47442038228897, −13.18651442212157, −12.39767575200028, −12.02802163441195, −11.56170953400466, −11.37957989804839, −10.64860247770511, −10.07770051933726, −9.625938116525398, −9.249062175291452, −8.700384749708659, −8.341246773483795, −7.624408518107295, −7.116817850991959, −6.649096397899256, −6.184480805771326, −5.826639630793830, −4.892573442094459, −4.463300058312840, −4.097745928921850, −3.511000293163259, −2.572187250057506, −2.409467722591750, −1.360563668148612, −0.9868012569022102, 0,
0.9868012569022102, 1.360563668148612, 2.409467722591750, 2.572187250057506, 3.511000293163259, 4.097745928921850, 4.463300058312840, 4.892573442094459, 5.826639630793830, 6.184480805771326, 6.649096397899256, 7.116817850991959, 7.624408518107295, 8.341246773483795, 8.700384749708659, 9.249062175291452, 9.625938116525398, 10.07770051933726, 10.64860247770511, 11.37957989804839, 11.56170953400466, 12.02802163441195, 12.39767575200028, 13.18651442212157, 13.47442038228897
