| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s + 6·11-s − 14-s + 16-s + 17-s − 2·19-s − 2·20-s + 6·22-s − 25-s − 28-s + 4·29-s + 32-s + 34-s + 2·35-s + 8·37-s − 2·38-s − 2·40-s − 2·41-s − 4·43-s + 6·44-s + 8·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.80·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s + 1.27·22-s − 1/5·25-s − 0.188·28-s + 0.742·29-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 1.31·37-s − 0.324·38-s − 0.316·40-s − 0.312·41-s − 0.609·43-s + 0.904·44-s + 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.517867168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517867168\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.050668208539988291528513300065, −8.241482900063155290280130992338, −7.38307216470903280183365304509, −6.61198640915252007394786259785, −6.07577312270787812557481081937, −4.91252524862077574990292793780, −3.94475495379555862756542826222, −3.68234594137598270185007075450, −2.39321496263075636114052800531, −0.977095425235087442309045097961,
0.977095425235087442309045097961, 2.39321496263075636114052800531, 3.68234594137598270185007075450, 3.94475495379555862756542826222, 4.91252524862077574990292793780, 6.07577312270787812557481081937, 6.61198640915252007394786259785, 7.38307216470903280183365304509, 8.241482900063155290280130992338, 9.050668208539988291528513300065
