| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s − 13-s + 15-s + 6·17-s − 8·19-s − 21-s + 4·23-s + 25-s − 27-s − 2·29-s + 4·31-s + 4·33-s − 35-s + 10·37-s + 39-s − 2·41-s − 8·43-s − 45-s − 8·47-s + 49-s − 6·51-s + 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s − 1.83·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s − 0.169·35-s + 1.64·37-s + 0.160·39-s − 0.312·41-s − 1.21·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.75299552596513, −15.19010328249659, −14.92855367128418, −14.32782873700447, −13.51763146223314, −12.92193375581025, −12.63831108118567, −11.98902928882785, −11.39276710001992, −10.91735140736048, −10.37341509550363, −9.932238838776741, −9.204365638718424, −8.264078162487634, −8.052547274508253, −7.460135529355257, −6.692482354522563, −6.140423401196292, −5.368961464907554, −4.888907134439819, −4.348028856148983, −3.464202944810000, −2.746082812300304, −1.929473741124518, −0.9166153405684555, 0,
0.9166153405684555, 1.929473741124518, 2.746082812300304, 3.464202944810000, 4.348028856148983, 4.888907134439819, 5.368961464907554, 6.140423401196292, 6.692482354522563, 7.460135529355257, 8.052547274508253, 8.264078162487634, 9.204365638718424, 9.932238838776741, 10.37341509550363, 10.91735140736048, 11.39276710001992, 11.98902928882785, 12.63831108118567, 12.92193375581025, 13.51763146223314, 14.32782873700447, 14.92855367128418, 15.19010328249659, 15.75299552596513
