| L(s) = 1 | − 7-s − 3·9-s − 2·11-s − 17-s + 6·19-s − 8·37-s + 6·41-s − 12·43-s + 4·47-s + 49-s − 6·53-s − 6·59-s + 2·61-s + 3·63-s − 8·67-s + 6·73-s + 2·77-s + 9·81-s − 6·83-s − 6·89-s + 2·97-s + 6·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.603·11-s − 0.242·17-s + 1.37·19-s − 1.31·37-s + 0.937·41-s − 1.82·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s − 0.781·59-s + 0.256·61-s + 0.377·63-s − 0.977·67-s + 0.702·73-s + 0.227·77-s + 81-s − 0.658·83-s − 0.635·89-s + 0.203·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.53872968786597, −12.81056250686046, −12.42829466364855, −11.87519660219471, −11.52185721091268, −11.01103220683265, −10.56186007810434, −9.995693951898751, −9.611904869363243, −9.019860650792982, −8.643727979191595, −8.104465419925362, −7.603362375716705, −7.130872486825238, −6.586262685039891, −5.951259088105794, −5.608664897413641, −5.031961788917433, −4.638028626258113, −3.765358558195903, −3.190377779552583, −2.965741696851278, −2.197042788366103, −1.556189321771296, −0.6700104553576071, 0,
0.6700104553576071, 1.556189321771296, 2.197042788366103, 2.965741696851278, 3.190377779552583, 3.765358558195903, 4.638028626258113, 5.031961788917433, 5.608664897413641, 5.951259088105794, 6.586262685039891, 7.130872486825238, 7.603362375716705, 8.104465419925362, 8.643727979191595, 9.019860650792982, 9.611904869363243, 9.995693951898751, 10.56186007810434, 11.01103220683265, 11.52185721091268, 11.87519660219471, 12.42829466364855, 12.81056250686046, 13.53872968786597
