| L(s) = 1 | + 11-s + 2·13-s − 6·17-s − 4·19-s − 6·29-s − 8·31-s − 6·37-s − 10·41-s + 4·43-s − 8·47-s − 7·49-s − 10·53-s + 12·59-s + 6·61-s + 4·67-s + 14·73-s + 4·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 49-s − 1.37·53-s + 1.56·59-s + 0.768·61-s + 0.488·67-s + 1.63·73-s + 0.439·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(\approx\) |
\(1.301112177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301112177\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.68102737288901, −15.15245826719492, −14.58005854113298, −14.13259457724228, −13.33605783891535, −12.97787345287306, −12.58999362506765, −11.65004598332175, −11.21023712284810, −10.87299106876480, −10.11242004888680, −9.493281978944194, −8.812225602386578, −8.543062683174690, −7.770703057130849, −6.971329616377915, −6.576557696984075, −5.967570613795054, −5.128703966597867, −4.613009666370539, −3.691884203203363, −3.407716080744818, −2.000238080552597, −1.902407496866484, −0.4495850770346294,
0.4495850770346294, 1.902407496866484, 2.000238080552597, 3.407716080744818, 3.691884203203363, 4.613009666370539, 5.128703966597867, 5.967570613795054, 6.576557696984075, 6.971329616377915, 7.770703057130849, 8.543062683174690, 8.812225602386578, 9.493281978944194, 10.11242004888680, 10.87299106876480, 11.21023712284810, 11.65004598332175, 12.58999362506765, 12.97787345287306, 13.33605783891535, 14.13259457724228, 14.58005854113298, 15.15245826719492, 15.68102737288901
