| L(s) = 1 | − 2·7-s − 4·13-s − 6·17-s − 4·19-s − 6·23-s − 5·25-s − 6·29-s + 8·31-s + 10·37-s + 6·41-s + 8·43-s + 6·47-s − 3·49-s + 8·61-s + 4·67-s − 6·71-s − 2·73-s − 14·79-s + 12·83-s + 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 1.25·23-s − 25-s − 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s + 1.02·61-s + 0.488·67-s − 0.712·71-s − 0.234·73-s − 1.57·79-s + 1.31·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.50519860595836, −13.85038951891626, −13.25082472628155, −12.98004326389451, −12.48637797956932, −11.81836846986971, −11.51840853709193, −10.80903591600239, −10.34882020059256, −9.731210202420295, −9.424143950861452, −8.918602985402226, −8.142963290823511, −7.759383587107171, −7.159695652149537, −6.541388039168566, −6.064681687161160, −5.691383343067341, −4.724403472761299, −4.206669142054927, −3.945532582959703, −2.873878903310139, −2.354849836450472, −1.974037948713719, −0.6824024615033998, 0,
0.6824024615033998, 1.974037948713719, 2.354849836450472, 2.873878903310139, 3.945532582959703, 4.206669142054927, 4.724403472761299, 5.691383343067341, 6.064681687161160, 6.541388039168566, 7.159695652149537, 7.759383587107171, 8.142963290823511, 8.918602985402226, 9.424143950861452, 9.731210202420295, 10.34882020059256, 10.80903591600239, 11.51840853709193, 11.81836846986971, 12.48637797956932, 12.98004326389451, 13.25082472628155, 13.85038951891626, 14.50519860595836
