| L(s) = 1 | − 4·7-s + 2·11-s + 13-s − 4·17-s − 2·19-s − 6·23-s + 2·29-s + 4·31-s − 6·37-s + 6·41-s − 8·43-s + 8·47-s + 9·49-s − 10·53-s − 14·59-s + 10·61-s − 4·67-s + 8·71-s + 10·73-s − 8·77-s + 8·79-s + 12·83-s + 18·89-s − 4·91-s + 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.458·19-s − 1.25·23-s + 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s − 1.82·59-s + 1.28·61-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.911·77-s + 0.900·79-s + 1.31·83-s + 1.90·89-s − 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 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.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.99506682639868, −14.11460917146439, −13.85637600157883, −13.34713088148632, −12.76632888491827, −12.30344947994079, −11.93402680778117, −11.20092852290190, −10.63192136292347, −10.17050527016246, −9.584999116055300, −9.164447824314577, −8.659398319773836, −8.021422326448310, −7.371127574780059, −6.542661427260019, −6.411696422030291, −5.990756565334416, −5.006310075487603, −4.426923758225721, −3.665757901087476, −3.391015810086760, −2.449876756732225, −1.917849228666232, −0.8013509446578811, 0,
0.8013509446578811, 1.917849228666232, 2.449876756732225, 3.391015810086760, 3.665757901087476, 4.426923758225721, 5.006310075487603, 5.990756565334416, 6.411696422030291, 6.542661427260019, 7.371127574780059, 8.021422326448310, 8.659398319773836, 9.164447824314577, 9.584999116055300, 10.17050527016246, 10.63192136292347, 11.20092852290190, 11.93402680778117, 12.30344947994079, 12.76632888491827, 13.34713088148632, 13.85637600157883, 14.11460917146439, 14.99506682639868
