| L(s) = 1 | − 4·7-s + 6·11-s − 13-s − 6·17-s − 5·19-s − 6·23-s + 9·29-s + 4·31-s − 2·37-s − 9·41-s + 2·43-s − 9·47-s + 9·49-s + 9·53-s + 8·61-s − 13·67-s + 4·73-s − 24·77-s + 16·79-s + 6·83-s + 15·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.80·11-s − 0.277·13-s − 1.45·17-s − 1.14·19-s − 1.25·23-s + 1.67·29-s + 0.718·31-s − 0.328·37-s − 1.40·41-s + 0.304·43-s − 1.31·47-s + 9/7·49-s + 1.23·53-s + 1.02·61-s − 1.58·67-s + 0.468·73-s − 2.73·77-s + 1.80·79-s + 0.658·83-s + 1.58·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.68735974505145, −13.24135178441073, −12.60871387452448, −12.19045120139147, −11.84503126805971, −11.41961767877783, −10.55952057440732, −10.29932555857484, −9.767888877979919, −9.316609493440515, −8.790607445810612, −8.505329490715065, −7.856999313601329, −6.876466809853879, −6.650539863388253, −6.428606802926654, −6.009144921201053, −5.039819211495831, −4.472434482309896, −3.941907001933028, −3.613459154081345, −2.827013955446967, −2.258313714338138, −1.629786310854005, −0.6969337561441926, 0,
0.6969337561441926, 1.629786310854005, 2.258313714338138, 2.827013955446967, 3.613459154081345, 3.941907001933028, 4.472434482309896, 5.039819211495831, 6.009144921201053, 6.428606802926654, 6.650539863388253, 6.876466809853879, 7.856999313601329, 8.505329490715065, 8.790607445810612, 9.316609493440515, 9.767888877979919, 10.29932555857484, 10.55952057440732, 11.41961767877783, 11.84503126805971, 12.19045120139147, 12.60871387452448, 13.24135178441073, 13.68735974505145
