| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s − 4·13-s − 2·15-s + 6·19-s − 21-s + 23-s − 25-s − 27-s + 6·29-s + 10·31-s + 2·35-s + 6·37-s + 4·39-s − 2·41-s + 12·43-s + 2·45-s − 10·47-s + 49-s + 10·53-s − 6·57-s − 12·59-s + 14·61-s + 63-s − 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 1.37·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.338·35-s + 0.986·37-s + 0.640·39-s − 0.312·41-s + 1.82·43-s + 0.298·45-s − 1.45·47-s + 1/7·49-s + 1.37·53-s − 0.794·57-s − 1.56·59-s + 1.79·61-s + 0.125·63-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.687304736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.687304736\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−15.22398966297961, −14.38168236402920, −14.04769989584290, −13.61825202547115, −12.97408903039033, −12.37015229162099, −11.87222297200218, −11.49939445754561, −10.80248714412314, −10.11622788491750, −9.816181090792265, −9.400897153400539, −8.595876031050859, −7.872030342750887, −7.449030770835865, −6.704630709276035, −6.189681495986510, −5.607390310267763, −4.927925463424645, −4.683109941253797, −3.730432317229636, −2.720076233142973, −2.360614215205117, −1.313250954845261, −0.7101791279007846,
0.7101791279007846, 1.313250954845261, 2.360614215205117, 2.720076233142973, 3.730432317229636, 4.683109941253797, 4.927925463424645, 5.607390310267763, 6.189681495986510, 6.704630709276035, 7.449030770835865, 7.872030342750887, 8.595876031050859, 9.400897153400539, 9.816181090792265, 10.11622788491750, 10.80248714412314, 11.49939445754561, 11.87222297200218, 12.37015229162099, 12.97408903039033, 13.61825202547115, 14.04769989584290, 14.38168236402920, 15.22398966297961
