| L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s + 11-s + 13-s + 2·15-s + 4·17-s + 6·19-s + 4·21-s + 6·23-s − 25-s + 27-s − 6·29-s + 6·31-s + 33-s + 8·35-s + 4·37-s + 39-s − 10·41-s + 8·43-s + 2·45-s − 12·47-s + 9·49-s + 4·51-s + 12·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.516·15-s + 0.970·17-s + 1.37·19-s + 0.872·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.07·31-s + 0.174·33-s + 1.35·35-s + 0.657·37-s + 0.160·39-s − 1.56·41-s + 1.21·43-s + 0.298·45-s − 1.75·47-s + 9/7·49-s + 0.560·51-s + 1.64·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.770297990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.770297990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.06885983307017, −14.72089973685203, −14.12114801284517, −13.67552708456323, −13.45315033025820, −12.62416694838099, −11.89683588206878, −11.58225580852287, −10.89324636701035, −10.38460748879483, −9.589915256190793, −9.403259309411725, −8.654215519457459, −8.086287603441485, −7.573453414459158, −7.116839268058787, −6.158680769499765, −5.648249686134968, −4.976235564516577, −4.592417941670890, −3.558377332812355, −3.051954011347022, −2.150555606685879, −1.474009133747539, −1.040586630174923,
1.040586630174923, 1.474009133747539, 2.150555606685879, 3.051954011347022, 3.558377332812355, 4.592417941670890, 4.976235564516577, 5.648249686134968, 6.158680769499765, 7.116839268058787, 7.573453414459158, 8.086287603441485, 8.654215519457459, 9.403259309411725, 9.589915256190793, 10.38460748879483, 10.89324636701035, 11.58225580852287, 11.89683588206878, 12.62416694838099, 13.45315033025820, 13.67552708456323, 14.12114801284517, 14.72089973685203, 15.06885983307017
