| L(s) = 1 | − 3-s − 4·5-s + 9-s + 13-s + 4·15-s − 2·17-s + 2·19-s + 5·23-s + 11·25-s − 27-s − 9·29-s + 3·31-s − 10·37-s − 39-s + 3·41-s + 43-s − 4·45-s + 6·47-s + 2·51-s − 2·53-s − 2·57-s − 7·59-s − 11·61-s − 4·65-s − 7·67-s − 5·69-s − 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 0.277·13-s + 1.03·15-s − 0.485·17-s + 0.458·19-s + 1.04·23-s + 11/5·25-s − 0.192·27-s − 1.67·29-s + 0.538·31-s − 1.64·37-s − 0.160·39-s + 0.468·41-s + 0.152·43-s − 0.596·45-s + 0.875·47-s + 0.280·51-s − 0.274·53-s − 0.264·57-s − 0.911·59-s − 1.40·61-s − 0.496·65-s − 0.855·67-s − 0.601·69-s − 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3013780203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3013780203\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.32909400358555, −12.64275845645008, −12.46955228595838, −11.85642467106507, −11.52144605671328, −10.97641449781321, −10.82913521629267, −10.22035486534933, −9.400263857720941, −8.975862295259438, −8.566363379962645, −7.867781881128124, −7.513134600982067, −7.053563051873954, −6.684789490929719, −5.862343888518817, −5.405964138281087, −4.739861659991307, −4.318583770684874, −3.827604243781553, −3.236804179127228, −2.771128095713361, −1.684860844257928, −1.076570104899420, −0.1945002393704216,
0.1945002393704216, 1.076570104899420, 1.684860844257928, 2.771128095713361, 3.236804179127228, 3.827604243781553, 4.318583770684874, 4.739861659991307, 5.405964138281087, 5.862343888518817, 6.684789490929719, 7.053563051873954, 7.513134600982067, 7.867781881128124, 8.566363379962645, 8.975862295259438, 9.400263857720941, 10.22035486534933, 10.82913521629267, 10.97641449781321, 11.52144605671328, 11.85642467106507, 12.46955228595838, 12.64275845645008, 13.32909400358555
