| L(s) = 1 | + 5-s − 2·7-s + 3·11-s − 13-s − 3·17-s − 8·19-s − 3·23-s + 25-s − 9·29-s + 7·31-s − 2·35-s + 2·37-s − 12·41-s + 7·43-s + 3·47-s − 3·49-s + 12·53-s + 3·55-s − 12·59-s − 10·61-s − 65-s + 4·67-s + 2·73-s − 6·77-s + 79-s − 18·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.904·11-s − 0.277·13-s − 0.727·17-s − 1.83·19-s − 0.625·23-s + 1/5·25-s − 1.67·29-s + 1.25·31-s − 0.338·35-s + 0.328·37-s − 1.87·41-s + 1.06·43-s + 0.437·47-s − 3/7·49-s + 1.64·53-s + 0.404·55-s − 1.56·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s + 0.234·73-s − 0.683·77-s + 0.112·79-s − 1.97·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−8.874969032343562701447767661632, −7.987431088496615763760094826771, −6.86599692706255249071269754257, −6.42202423051288271685955538968, −5.70096124404004576026267487667, −4.48286975641059636887413384127, −3.84298744888411936217576546090, −2.64315915730396133980270708555, −1.71464041150170530329847422571, 0,
1.71464041150170530329847422571, 2.64315915730396133980270708555, 3.84298744888411936217576546090, 4.48286975641059636887413384127, 5.70096124404004576026267487667, 6.42202423051288271685955538968, 6.86599692706255249071269754257, 7.987431088496615763760094826771, 8.874969032343562701447767661632
