| L(s) = 1 | + 3-s − 5-s + 9-s − 6·13-s − 15-s + 3·17-s − 3·23-s + 25-s + 27-s − 6·29-s − 7·31-s + 8·37-s − 6·39-s − 2·41-s + 10·43-s − 45-s + 3·47-s − 7·49-s + 3·51-s − 3·53-s + 2·59-s + 7·61-s + 6·65-s + 12·67-s − 3·69-s − 12·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.727·17-s − 0.625·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.25·31-s + 1.31·37-s − 0.960·39-s − 0.312·41-s + 1.52·43-s − 0.149·45-s + 0.437·47-s − 49-s + 0.420·51-s − 0.412·53-s + 0.260·59-s + 0.896·61-s + 0.744·65-s + 1.46·67-s − 0.361·69-s − 1.42·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.20051858697425, −13.13255840478991, −12.88703323846050, −12.52250338188954, −11.92687695220894, −11.43204824748463, −11.04375957610048, −10.24646597698998, −9.892578425858630, −9.489556988030166, −8.961843424634702, −8.408370811164663, −7.739790299324206, −7.365903192469437, −7.285298902925435, −6.289364701994587, −5.779442799042002, −5.154969114576829, −4.639464640846024, −4.013306219274530, −3.552005003641930, −2.873669276901667, −2.296806169844113, −1.774158119658825, −0.7976008567857180, 0,
0.7976008567857180, 1.774158119658825, 2.296806169844113, 2.873669276901667, 3.552005003641930, 4.013306219274530, 4.639464640846024, 5.154969114576829, 5.779442799042002, 6.289364701994587, 7.285298902925435, 7.365903192469437, 7.739790299324206, 8.408370811164663, 8.961843424634702, 9.489556988030166, 9.892578425858630, 10.24646597698998, 11.04375957610048, 11.43204824748463, 11.92687695220894, 12.52250338188954, 12.88703323846050, 13.13255840478991, 14.20051858697425
