| L(s) = 1 | + 3-s − 3·7-s − 2·9-s − 5·11-s − 4·13-s + 2·17-s + 6·19-s − 3·21-s − 3·23-s − 5·27-s − 2·29-s − 5·33-s + 6·37-s − 4·39-s + 7·41-s − 10·43-s − 4·47-s + 2·49-s + 2·51-s − 6·53-s + 6·57-s + 3·59-s + 5·61-s + 6·63-s + 8·67-s − 3·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s − 2/3·9-s − 1.50·11-s − 1.10·13-s + 0.485·17-s + 1.37·19-s − 0.654·21-s − 0.625·23-s − 0.962·27-s − 0.371·29-s − 0.870·33-s + 0.986·37-s − 0.640·39-s + 1.09·41-s − 1.52·43-s − 0.583·47-s + 2/7·49-s + 0.280·51-s − 0.824·53-s + 0.794·57-s + 0.390·59-s + 0.640·61-s + 0.755·63-s + 0.977·67-s − 0.361·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.67069408875956, −13.15634162728152, −12.89395059758855, −12.36713742043438, −11.79530802131176, −11.36579844622820, −10.79527708511246, −10.03616443051165, −9.725353680759459, −9.633675148482012, −8.859251375734213, −8.162119426369369, −7.907057821999232, −7.391538925521877, −6.935239142333473, −6.146159148516450, −5.690517710913582, −5.238999067841173, −4.725530773783205, −3.860267608625684, −3.249687489185634, −2.878032358711043, −2.500697984844434, −1.754015183531456, −0.6164605175243276, 0,
0.6164605175243276, 1.754015183531456, 2.500697984844434, 2.878032358711043, 3.249687489185634, 3.860267608625684, 4.725530773783205, 5.238999067841173, 5.690517710913582, 6.146159148516450, 6.935239142333473, 7.391538925521877, 7.907057821999232, 8.162119426369369, 8.859251375734213, 9.633675148482012, 9.725353680759459, 10.03616443051165, 10.79527708511246, 11.36579844622820, 11.79530802131176, 12.36713742043438, 12.89395059758855, 13.15634162728152, 13.67069408875956
