| L(s) = 1 | − 2·5-s − 7-s + 2·11-s + 2·13-s − 6·17-s − 6·19-s + 6·23-s − 25-s − 9·29-s + 2·31-s + 2·35-s − 9·37-s + 10·41-s + 2·43-s − 10·47-s − 6·49-s + 8·53-s − 4·55-s + 3·59-s + 6·61-s − 4·65-s − 10·67-s + 5·71-s − 9·73-s − 2·77-s + 12·79-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s + 0.554·13-s − 1.45·17-s − 1.37·19-s + 1.25·23-s − 1/5·25-s − 1.67·29-s + 0.359·31-s + 0.338·35-s − 1.47·37-s + 1.56·41-s + 0.304·43-s − 1.45·47-s − 6/7·49-s + 1.09·53-s − 0.539·55-s + 0.390·59-s + 0.768·61-s − 0.496·65-s − 1.22·67-s + 0.593·71-s − 1.05·73-s − 0.227·77-s + 1.35·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.17474858946130, −12.72158172022536, −12.33507028632506, −11.52520472986890, −11.41255069663669, −10.92835538956348, −10.56072181692721, −9.889247239745159, −9.243663254897408, −8.916556592907268, −8.596107068614853, −7.993608026517822, −7.475737794032790, −6.827931829412187, −6.683741619606307, −6.054988358466189, −5.517091150484460, −4.741743046537844, −4.369388745882348, −3.769994933318268, −3.545894676119887, −2.737527962301486, −2.085476887836704, −1.546329183747091, −0.6035815796703914, 0,
0.6035815796703914, 1.546329183747091, 2.085476887836704, 2.737527962301486, 3.545894676119887, 3.769994933318268, 4.369388745882348, 4.741743046537844, 5.517091150484460, 6.054988358466189, 6.683741619606307, 6.827931829412187, 7.475737794032790, 7.993608026517822, 8.596107068614853, 8.916556592907268, 9.243663254897408, 9.889247239745159, 10.56072181692721, 10.92835538956348, 11.41255069663669, 11.52520472986890, 12.33507028632506, 12.72158172022536, 13.17474858946130
