| L(s) = 1 | − 4·7-s − 11-s − 2·13-s − 2·17-s + 4·19-s + 2·29-s + 2·37-s + 6·41-s + 8·47-s + 9·49-s − 2·53-s − 4·59-s − 10·61-s − 4·67-s − 14·73-s + 4·77-s + 16·79-s − 10·89-s + 8·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.371·29-s + 0.328·37-s + 0.937·41-s + 1.16·47-s + 9/7·49-s − 0.274·53-s − 0.520·59-s − 1.28·61-s − 0.488·67-s − 1.63·73-s + 0.455·77-s + 1.80·79-s − 1.05·89-s + 0.838·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.94146699168799, −15.50824959814220, −14.99897719257288, −14.16844224405989, −13.77310320532213, −13.14230090753811, −12.72845215671929, −12.16662423069177, −11.68549598495120, −10.83441655673006, −10.38606208556640, −9.747992698366804, −9.308384995535103, −8.857684996502494, −7.922493516233171, −7.366213903966423, −6.870037655948136, −6.099957424395163, −5.753502959770234, −4.822853411060116, −4.231045216970585, −3.314124598677663, −2.900864469770616, −2.134058965000547, −0.9133980761383935, 0,
0.9133980761383935, 2.134058965000547, 2.900864469770616, 3.314124598677663, 4.231045216970585, 4.822853411060116, 5.753502959770234, 6.099957424395163, 6.870037655948136, 7.366213903966423, 7.922493516233171, 8.857684996502494, 9.308384995535103, 9.747992698366804, 10.38606208556640, 10.83441655673006, 11.68549598495120, 12.16662423069177, 12.72845215671929, 13.14230090753811, 13.77310320532213, 14.16844224405989, 14.99897719257288, 15.50824959814220, 15.94146699168799
