| L(s) = 1 | − 3·7-s + 3·11-s − 4·13-s − 4·17-s − 2·29-s − 3·31-s + 8·41-s − 12·43-s − 12·47-s + 2·49-s + 5·53-s + 12·67-s + 12·71-s − 9·73-s − 9·77-s − 12·79-s + 9·83-s + 4·89-s + 12·91-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 12·119-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.904·11-s − 1.10·13-s − 0.970·17-s − 0.371·29-s − 0.538·31-s + 1.24·41-s − 1.82·43-s − 1.75·47-s + 2/7·49-s + 0.686·53-s + 1.46·67-s + 1.42·71-s − 1.05·73-s − 1.02·77-s − 1.35·79-s + 0.987·83-s + 0.423·89-s + 1.25·91-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.10·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.25721662240711, −12.98595340183477, −12.60837584918685, −11.98433131122787, −11.62121329880104, −11.13908116031824, −10.58223293166860, −9.933421167167229, −9.597626142554660, −9.352114915154169, −8.677085256107362, −8.266299350397039, −7.523844477004327, −7.034325689128872, −6.648690341342947, −6.272682634772265, −5.648315611625166, −4.971849151097069, −4.562888840526325, −3.831205921066911, −3.455284610462865, −2.817233888276475, −2.175464277163508, −1.643220900127772, −0.6407750064018201, 0,
0.6407750064018201, 1.643220900127772, 2.175464277163508, 2.817233888276475, 3.455284610462865, 3.831205921066911, 4.562888840526325, 4.971849151097069, 5.648315611625166, 6.272682634772265, 6.648690341342947, 7.034325689128872, 7.523844477004327, 8.266299350397039, 8.677085256107362, 9.352114915154169, 9.597626142554660, 9.933421167167229, 10.58223293166860, 11.13908116031824, 11.62121329880104, 11.98433131122787, 12.60837584918685, 12.98595340183477, 13.25721662240711
