| L(s) = 1 | + 5-s + 2·13-s + 4·19-s + 25-s + 6·29-s + 2·31-s − 6·37-s − 4·41-s − 4·43-s − 8·47-s − 2·53-s + 4·59-s + 2·61-s + 2·65-s − 8·67-s − 8·71-s + 6·73-s + 6·79-s + 4·83-s + 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.554·13-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.359·31-s − 0.986·37-s − 0.624·41-s − 0.609·43-s − 1.16·47-s − 0.274·53-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 0.949·71-s + 0.702·73-s + 0.675·79-s + 0.439·83-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.13888058493554, −12.47761622760342, −11.89179182932800, −11.78789124244032, −11.14358980143162, −10.51034265187305, −10.32813654090793, −9.736145381153139, −9.331130531683825, −8.814463818607421, −8.291219615778356, −7.997975721794003, −7.307922224866821, −6.724973011274464, −6.518065452817106, −5.817808224782654, −5.419232467467981, −4.816515296861335, −4.501068104274681, −3.564701795428758, −3.336678395511840, −2.704187639130293, −2.016358416704345, −1.427250996920427, −0.9012168883101616, 0,
0.9012168883101616, 1.427250996920427, 2.016358416704345, 2.704187639130293, 3.336678395511840, 3.564701795428758, 4.501068104274681, 4.816515296861335, 5.419232467467981, 5.817808224782654, 6.518065452817106, 6.724973011274464, 7.307922224866821, 7.997975721794003, 8.291219615778356, 8.814463818607421, 9.331130531683825, 9.736145381153139, 10.32813654090793, 10.51034265187305, 11.14358980143162, 11.78789124244032, 11.89179182932800, 12.47761622760342, 13.13888058493554
