| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s + 18-s − 7·19-s − 20-s − 21-s + 3·23-s − 24-s − 4·25-s − 26-s − 27-s + 28-s + 3·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.60·19-s − 0.223·20-s − 0.218·21-s + 0.625·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.360858106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360858106\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.20630465200613, −13.60821273853202, −13.24062236159570, −12.57155392242398, −12.22785415192488, −11.75262814537946, −11.26402985412837, −10.86538238345709, −10.19858428882483, −9.912741378107506, −9.038609398457030, −8.395590836095821, −7.964246861307720, −7.432633221606735, −6.692553630868033, −6.352324519648441, −5.843206398453806, −4.964938777959108, −4.751795591625757, −4.149981488401613, −3.559807692412961, −2.775015674268798, −2.143312803538977, −1.370447152230187, −0.4809560011511493,
0.4809560011511493, 1.370447152230187, 2.143312803538977, 2.775015674268798, 3.559807692412961, 4.149981488401613, 4.751795591625757, 4.964938777959108, 5.843206398453806, 6.352324519648441, 6.692553630868033, 7.432633221606735, 7.964246861307720, 8.395590836095821, 9.038609398457030, 9.912741378107506, 10.19858428882483, 10.86538238345709, 11.26402985412837, 11.75262814537946, 12.22785415192488, 12.57155392242398, 13.24062236159570, 13.60821273853202, 14.20630465200613
