| L(s) = 1 | + 3·3-s − 5-s − 7-s + 6·9-s + 4·13-s − 3·15-s − 2·17-s − 6·19-s − 3·21-s + 5·23-s − 4·25-s + 9·27-s − 10·29-s − 31-s + 35-s − 5·37-s + 12·39-s + 2·41-s − 8·43-s − 6·45-s − 8·47-s + 49-s − 6·51-s − 6·53-s − 18·57-s − 3·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 1.10·13-s − 0.774·15-s − 0.485·17-s − 1.37·19-s − 0.654·21-s + 1.04·23-s − 4/5·25-s + 1.73·27-s − 1.85·29-s − 0.179·31-s + 0.169·35-s − 0.821·37-s + 1.92·39-s + 0.312·41-s − 1.21·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s − 2.38·57-s − 0.390·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 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.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−16.31708239509841, −15.63005574598782, −15.33397270611150, −14.79886822500248, −14.38024309793918, −13.53177843017387, −13.15571210090187, −12.95801208047788, −12.07626524103951, −11.17859033687482, −10.83392425733800, −9.998496611960822, −9.340965711176791, −8.941137724366696, −8.349067464793441, −7.963592896581777, −7.177829296022279, −6.668397066703540, −5.892147342895214, −4.843347312845342, −4.028984289515527, −3.608976401511123, −3.054671355646993, −2.088595514830599, −1.549179836342091, 0,
1.549179836342091, 2.088595514830599, 3.054671355646993, 3.608976401511123, 4.028984289515527, 4.843347312845342, 5.892147342895214, 6.668397066703540, 7.177829296022279, 7.963592896581777, 8.349067464793441, 8.941137724366696, 9.340965711176791, 9.998496611960822, 10.83392425733800, 11.17859033687482, 12.07626524103951, 12.95801208047788, 13.15571210090187, 13.53177843017387, 14.38024309793918, 14.79886822500248, 15.33397270611150, 15.63005574598782, 16.31708239509841
