| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 2·11-s + 12-s − 13-s + 16-s + 3·17-s − 18-s − 2·22-s + 23-s − 24-s + 26-s + 27-s − 5·29-s − 7·31-s − 32-s + 2·33-s − 3·34-s + 36-s + 2·37-s − 39-s − 7·41-s + 11·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.928·29-s − 1.25·31-s − 0.176·32-s + 0.348·33-s − 0.514·34-s + 1/6·36-s + 0.328·37-s − 0.160·39-s − 1.09·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877819265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877819265\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.953715794168135685691444613385, −7.29463165599210752963395910201, −6.83314880184316849750110457530, −5.84949340099089637536000881686, −5.23310040607092510124728352755, −4.06679881044300376346327607010, −3.52034780939110352871062506538, −2.54181361721711325909080024430, −1.77270023056846991043255622018, −0.75991560168984159160120946243,
0.75991560168984159160120946243, 1.77270023056846991043255622018, 2.54181361721711325909080024430, 3.52034780939110352871062506538, 4.06679881044300376346327607010, 5.23310040607092510124728352755, 5.84949340099089637536000881686, 6.83314880184316849750110457530, 7.29463165599210752963395910201, 7.953715794168135685691444613385
