| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 5·11-s + 5·13-s + 15-s − 7·19-s − 2·21-s + 3·23-s − 4·25-s − 27-s + 2·29-s − 2·31-s − 5·33-s − 2·35-s − 8·37-s − 5·39-s + 7·41-s − 5·43-s − 45-s − 8·47-s − 3·49-s − 4·53-s − 5·55-s + 7·57-s + 14·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s + 0.258·15-s − 1.60·19-s − 0.436·21-s + 0.625·23-s − 4/5·25-s − 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.870·33-s − 0.338·35-s − 1.31·37-s − 0.800·39-s + 1.09·41-s − 0.762·43-s − 0.149·45-s − 1.16·47-s − 3/7·49-s − 0.549·53-s − 0.674·55-s + 0.927·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.73027884988739, −14.19420253089629, −13.69840790513416, −12.99983082200862, −12.61462308865591, −12.02341699514025, −11.39582242249178, −11.20110081294381, −10.86881625194670, −10.03653945001605, −9.561536590810441, −8.719398151980262, −8.494579079081250, −8.045499283352446, −7.124577958357988, −6.660970230472950, −6.309965208926891, −5.622861893884817, −4.984580152868217, −4.301791389401078, −3.862499147856973, −3.448696181067093, −2.241410346741991, −1.561186637898812, −1.052851979309691, 0,
1.052851979309691, 1.561186637898812, 2.241410346741991, 3.448696181067093, 3.862499147856973, 4.301791389401078, 4.984580152868217, 5.622861893884817, 6.309965208926891, 6.660970230472950, 7.124577958357988, 8.045499283352446, 8.494579079081250, 8.719398151980262, 9.561536590810441, 10.03653945001605, 10.86881625194670, 11.20110081294381, 11.39582242249178, 12.02341699514025, 12.61462308865591, 12.99983082200862, 13.69840790513416, 14.19420253089629, 14.73027884988739
