| L(s) = 1 | + 2·5-s − 2·7-s − 13-s − 2·17-s + 4·23-s − 25-s − 2·29-s − 2·31-s − 4·35-s − 6·37-s + 6·41-s + 2·43-s + 6·47-s − 3·49-s + 6·53-s + 8·59-s + 2·61-s − 2·65-s + 8·67-s − 2·71-s − 14·73-s − 4·79-s − 8·83-s − 4·85-s + 14·89-s + 2·91-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.277·13-s − 0.485·17-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 0.304·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.04·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s − 0.237·71-s − 1.63·73-s − 0.450·79-s − 0.878·83-s − 0.433·85-s + 1.48·89-s + 0.209·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.25137446930069, −15.87984762195084, −15.24959269476839, −14.55510512475000, −14.16675924608387, −13.32934020893515, −13.16166285531744, −12.56800623041216, −11.86100409703490, −11.24663860554410, −10.50104462766456, −10.14022967786801, −9.354025023828100, −9.134474382523241, −8.393320887159469, −7.489213311828245, −6.952300026498294, −6.380604802432898, −5.662768624151575, −5.249028617771603, −4.295616381161132, −3.608577930658963, −2.724204638009104, −2.166915124695014, −1.185159110513533, 0,
1.185159110513533, 2.166915124695014, 2.724204638009104, 3.608577930658963, 4.295616381161132, 5.249028617771603, 5.662768624151575, 6.380604802432898, 6.952300026498294, 7.489213311828245, 8.393320887159469, 9.134474382523241, 9.354025023828100, 10.14022967786801, 10.50104462766456, 11.24663860554410, 11.86100409703490, 12.56800623041216, 13.16166285531744, 13.32934020893515, 14.16675924608387, 14.55510512475000, 15.24959269476839, 15.87984762195084, 16.25137446930069
