| L(s) = 1 | − 5-s − 7-s + 3·11-s − 4·13-s + 17-s + 5·19-s − 6·23-s + 25-s − 3·29-s − 4·31-s + 35-s + 5·37-s + 9·41-s − 10·43-s − 3·47-s − 6·49-s + 3·53-s − 3·55-s + 12·59-s − 10·61-s + 4·65-s − 10·67-s − 73-s − 3·77-s − 4·79-s − 12·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.904·11-s − 1.10·13-s + 0.242·17-s + 1.14·19-s − 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.718·31-s + 0.169·35-s + 0.821·37-s + 1.40·41-s − 1.52·43-s − 0.437·47-s − 6/7·49-s + 0.412·53-s − 0.404·55-s + 1.56·59-s − 1.28·61-s + 0.496·65-s − 1.22·67-s − 0.117·73-s − 0.341·77-s − 0.450·79-s − 1.31·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−8.219288169141741569681195274145, −7.56001677968101544218010762238, −6.94897865497580997612971878899, −6.06828456918375522219020208549, −5.27739351736528468546333279040, −4.31866223394593001645605432652, −3.59992371204925941933436012016, −2.67504292094169190142703846281, −1.45040595831206287698471929946, 0,
1.45040595831206287698471929946, 2.67504292094169190142703846281, 3.59992371204925941933436012016, 4.31866223394593001645605432652, 5.27739351736528468546333279040, 6.06828456918375522219020208549, 6.94897865497580997612971878899, 7.56001677968101544218010762238, 8.219288169141741569681195274145
