| L(s) = 1 | − 7-s − 3·9-s − 11-s + 2·13-s − 2·17-s − 8·23-s + 2·29-s + 8·31-s − 2·37-s + 10·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 10·61-s + 3·63-s + 12·67-s − 16·71-s + 14·73-s + 77-s + 9·81-s − 6·89-s − 2·91-s − 10·97-s + 3·99-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.301·11-s + 0.554·13-s − 0.485·17-s − 1.66·23-s + 0.371·29-s + 1.43·31-s − 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s + 0.377·63-s + 1.46·67-s − 1.89·71-s + 1.63·73-s + 0.113·77-s + 81-s − 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.301·99-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281370738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281370738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.64694757163457, −13.16452866097857, −12.43008696464831, −12.12156461487859, −11.69348307199507, −11.01789228974530, −10.75858900180253, −10.09599994855953, −9.717115648309749, −9.073599512969937, −8.600083797814461, −8.172409635988761, −7.753952413101823, −7.040115058429453, −6.408608334493118, −6.055380684479780, −5.614460468102487, −4.981824482516064, −4.202558489818999, −3.915712062707340, −3.068925786624003, −2.609552474464280, −2.087422537693130, −1.137002454932159, −0.3661765325761320,
0.3661765325761320, 1.137002454932159, 2.087422537693130, 2.609552474464280, 3.068925786624003, 3.915712062707340, 4.202558489818999, 4.981824482516064, 5.614460468102487, 6.055380684479780, 6.408608334493118, 7.040115058429453, 7.753952413101823, 8.172409635988761, 8.600083797814461, 9.073599512969937, 9.717115648309749, 10.09599994855953, 10.75858900180253, 11.01789228974530, 11.69348307199507, 12.12156461487859, 12.43008696464831, 13.16452866097857, 13.64694757163457
