| L(s) = 1 | − 3·5-s + 2·7-s − 13-s + 17-s + 2·23-s + 4·25-s + 7·29-s − 4·31-s − 6·35-s − 3·37-s + 9·41-s + 6·43-s − 12·47-s − 3·49-s + 9·53-s − 6·59-s − 6·61-s + 3·65-s − 2·67-s + 10·71-s − 14·73-s − 8·79-s − 4·83-s − 3·85-s − 89-s − 2·91-s − 17·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s − 0.277·13-s + 0.242·17-s + 0.417·23-s + 4/5·25-s + 1.29·29-s − 0.718·31-s − 1.01·35-s − 0.493·37-s + 1.40·41-s + 0.914·43-s − 1.75·47-s − 3/7·49-s + 1.23·53-s − 0.781·59-s − 0.768·61-s + 0.372·65-s − 0.244·67-s + 1.18·71-s − 1.63·73-s − 0.900·79-s − 0.439·83-s − 0.325·85-s − 0.105·89-s − 0.209·91-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−15.21543387745993, −14.62405193097348, −14.39968690731647, −13.72745241163857, −13.00704903974300, −12.41941045345818, −12.10987148933634, −11.37766973238868, −11.22224295725876, −10.54476936679704, −9.975822069364238, −9.197129054144792, −8.680324412675365, −8.088301481919631, −7.709450756024096, −7.198565249079103, −6.594909168425666, −5.771550317970510, −5.121229861032481, −4.463768898497631, −4.139785589752447, −3.272201126001684, −2.752892576536923, −1.744000983903769, −0.9384761098811085, 0,
0.9384761098811085, 1.744000983903769, 2.752892576536923, 3.272201126001684, 4.139785589752447, 4.463768898497631, 5.121229861032481, 5.771550317970510, 6.594909168425666, 7.198565249079103, 7.709450756024096, 8.088301481919631, 8.680324412675365, 9.197129054144792, 9.975822069364238, 10.54476936679704, 11.22224295725876, 11.37766973238868, 12.10987148933634, 12.41941045345818, 13.00704903974300, 13.72745241163857, 14.39968690731647, 14.62405193097348, 15.21543387745993
