| L(s) = 1 | + 3-s + 3·5-s + 2·7-s − 2·9-s − 11-s + 3·15-s − 2·19-s + 2·21-s + 3·23-s + 4·25-s − 5·27-s + 6·29-s − 31-s − 33-s + 6·35-s − 7·37-s − 6·41-s + 8·43-s − 6·45-s + 12·47-s − 3·49-s + 6·53-s − 3·55-s − 2·57-s − 9·59-s − 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s + 0.774·15-s − 0.458·19-s + 0.436·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s + 1.11·29-s − 0.179·31-s − 0.174·33-s + 1.01·35-s − 1.15·37-s − 0.937·41-s + 1.21·43-s − 0.894·45-s + 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.404·55-s − 0.264·57-s − 1.17·59-s − 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.424717368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.424717368\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.60208252035900, −13.35933428918164, −12.70158389648640, −12.07253993067942, −11.77644918206403, −10.94055353635662, −10.57465727036971, −10.32920233710284, −9.498970283065333, −9.141551935661685, −8.726631448786774, −8.278673492746778, −7.700654661267469, −7.167106490432674, −6.461393987241804, −6.040800938412448, −5.436553561851270, −5.079586247514910, −4.483624111882092, −3.701746088557729, −3.039800738490409, −2.463005519283122, −2.058177441452159, −1.439110334640129, −0.5979808390706902,
0.5979808390706902, 1.439110334640129, 2.058177441452159, 2.463005519283122, 3.039800738490409, 3.701746088557729, 4.483624111882092, 5.079586247514910, 5.436553561851270, 6.040800938412448, 6.461393987241804, 7.167106490432674, 7.700654661267469, 8.278673492746778, 8.726631448786774, 9.141551935661685, 9.498970283065333, 10.32920233710284, 10.57465727036971, 10.94055353635662, 11.77644918206403, 12.07253993067942, 12.70158389648640, 13.35933428918164, 13.60208252035900
