| L(s) = 1 | − 7-s + 3·11-s − 2·13-s − 4·17-s − 2·19-s − 9·23-s + 7·29-s + 10·31-s − 5·37-s + 8·41-s + 7·43-s − 10·47-s + 49-s + 10·53-s + 6·59-s − 8·61-s + 13·67-s − 71-s − 4·73-s − 3·77-s − 13·79-s + 14·83-s − 6·89-s + 2·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.87·23-s + 1.29·29-s + 1.79·31-s − 0.821·37-s + 1.24·41-s + 1.06·43-s − 1.45·47-s + 1/7·49-s + 1.37·53-s + 0.781·59-s − 1.02·61-s + 1.58·67-s − 0.118·71-s − 0.468·73-s − 0.341·77-s − 1.46·79-s + 1.53·83-s − 0.635·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 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
−16.52595381624192, −15.88024932695551, −15.68014625660843, −14.82766994455921, −14.28443885160726, −13.83666328654438, −13.25936487662567, −12.48227131554172, −12.02940220876891, −11.62862210033418, −10.78783323056414, −10.16108869358914, −9.735923591139389, −9.031023953516067, −8.410244556328462, −7.891916372254802, −6.961467202754420, −6.464852088056912, −6.039297738958050, −5.058203608510465, −4.246713455665786, −3.942596037635273, −2.763951537158340, −2.232661176980796, −1.135796228869574, 0,
1.135796228869574, 2.232661176980796, 2.763951537158340, 3.942596037635273, 4.246713455665786, 5.058203608510465, 6.039297738958050, 6.464852088056912, 6.961467202754420, 7.891916372254802, 8.410244556328462, 9.031023953516067, 9.735923591139389, 10.16108869358914, 10.78783323056414, 11.62862210033418, 12.02940220876891, 12.48227131554172, 13.25936487662567, 13.83666328654438, 14.28443885160726, 14.82766994455921, 15.68014625660843, 15.88024932695551, 16.52595381624192
