| L(s) = 1 | − 2·7-s + 4·11-s + 13-s − 8·17-s − 2·19-s − 4·23-s − 8·29-s + 10·31-s − 6·37-s + 6·41-s + 8·43-s + 8·47-s − 3·49-s + 12·53-s − 4·59-s + 10·61-s − 2·67-s − 6·73-s − 8·77-s + 12·79-s + 4·83-s − 6·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.20·11-s + 0.277·13-s − 1.94·17-s − 0.458·19-s − 0.834·23-s − 1.48·29-s + 1.79·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s + 1.64·53-s − 0.520·59-s + 1.28·61-s − 0.244·67-s − 0.702·73-s − 0.911·77-s + 1.35·79-s + 0.439·83-s − 0.635·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.68192909822992, −15.30442141858397, −14.67971553098559, −13.98673494912340, −13.61184066626614, −13.05493994459882, −12.51655567782416, −11.89388955834313, −11.42872681417059, −10.81041324190584, −10.28255925117562, −9.568222478774334, −9.014941675499891, −8.765808211760327, −7.946318550898144, −7.144244829475990, −6.645400660934827, −6.198209182314357, −5.642988019289612, −4.590566268722672, −4.053995476384019, −3.665614412490217, −2.555285534642237, −2.076746863709567, −1.001088476619033, 0,
1.001088476619033, 2.076746863709567, 2.555285534642237, 3.665614412490217, 4.053995476384019, 4.590566268722672, 5.642988019289612, 6.198209182314357, 6.645400660934827, 7.144244829475990, 7.946318550898144, 8.765808211760327, 9.014941675499891, 9.568222478774334, 10.28255925117562, 10.81041324190584, 11.42872681417059, 11.89388955834313, 12.51655567782416, 13.05493994459882, 13.61184066626614, 13.98673494912340, 14.67971553098559, 15.30442141858397, 15.68192909822992
