| L(s) = 1 | − 2·3-s − 7-s + 9-s + 11-s + 4·13-s − 4·17-s + 2·21-s − 4·23-s + 4·27-s + 6·29-s − 10·31-s − 2·33-s − 6·37-s − 8·39-s + 4·41-s − 12·43-s − 10·47-s + 49-s + 8·51-s − 6·53-s + 2·59-s − 63-s − 8·67-s + 8·69-s + 12·71-s + 8·73-s − 77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.970·17-s + 0.436·21-s − 0.834·23-s + 0.769·27-s + 1.11·29-s − 1.79·31-s − 0.348·33-s − 0.986·37-s − 1.28·39-s + 0.624·41-s − 1.82·43-s − 1.45·47-s + 1/7·49-s + 1.12·51-s − 0.824·53-s + 0.260·59-s − 0.125·63-s − 0.977·67-s + 0.963·69-s + 1.42·71-s + 0.936·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.74267608839280, −13.14876635920536, −12.82141812491650, −12.26375389236192, −11.76421010491831, −11.36520949953828, −10.95738836393777, −10.53089075494458, −10.02360015150574, −9.440032749670727, −8.857018551845271, −8.436587937842144, −7.945598858069308, −7.084414550963911, −6.575522132074163, −6.422155328187809, −5.769982026032600, −5.319453902009408, −4.712751862918035, −4.185075210696150, −3.472410349630218, −3.091607546823458, −2.004997793301213, −1.582536905007839, −0.6420658748593552, 0,
0.6420658748593552, 1.582536905007839, 2.004997793301213, 3.091607546823458, 3.472410349630218, 4.185075210696150, 4.712751862918035, 5.319453902009408, 5.769982026032600, 6.422155328187809, 6.575522132074163, 7.084414550963911, 7.945598858069308, 8.436587937842144, 8.857018551845271, 9.440032749670727, 10.02360015150574, 10.53089075494458, 10.95738836393777, 11.36520949953828, 11.76421010491831, 12.26375389236192, 12.82141812491650, 13.14876635920536, 13.74267608839280
