| L(s) = 1 | + 3-s + 9-s − 11-s + 4·13-s + 8·19-s − 4·23-s + 27-s + 2·29-s − 4·31-s − 33-s + 8·37-s + 4·39-s − 2·41-s + 4·47-s − 7·49-s + 12·53-s + 8·57-s + 12·59-s + 2·61-s − 12·67-s − 4·69-s − 12·73-s − 12·79-s + 81-s + 8·83-s + 2·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.83·19-s − 0.834·23-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.583·47-s − 49-s + 1.64·53-s + 1.05·57-s + 1.56·59-s + 0.256·61-s − 1.46·67-s − 0.481·69-s − 1.40·73-s − 1.35·79-s + 1/9·81-s + 0.878·83-s + 0.214·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.086982422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.086982422\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−16.27880484521405, −15.77906405510440, −15.06306068351977, −14.53994197552355, −13.95320247690123, −13.37989018728709, −13.13690425539523, −12.23335670787889, −11.65965717971483, −11.20106585030395, −10.34131283098727, −9.932564618829147, −9.256168870815270, −8.661903192849723, −8.098304143393585, −7.466576342404264, −6.957597910555607, −5.961046971808921, −5.603991082495799, −4.669851792776054, −3.912094075223639, −3.305385238740401, −2.597461439887645, −1.636964854313727, −0.7994687688430399,
0.7994687688430399, 1.636964854313727, 2.597461439887645, 3.305385238740401, 3.912094075223639, 4.669851792776054, 5.603991082495799, 5.961046971808921, 6.957597910555607, 7.466576342404264, 8.098304143393585, 8.661903192849723, 9.256168870815270, 9.932564618829147, 10.34131283098727, 11.20106585030395, 11.65965717971483, 12.23335670787889, 13.13690425539523, 13.37989018728709, 13.95320247690123, 14.53994197552355, 15.06306068351977, 15.77906405510440, 16.27880484521405
