| L(s) = 1 | + 7-s + 3·11-s − 2·13-s − 3·17-s − 4·19-s − 6·29-s + 8·31-s − 2·37-s + 43-s + 6·47-s − 6·49-s − 9·53-s + 12·59-s − 61-s + 7·67-s + 15·71-s + 4·73-s + 3·77-s − 10·79-s − 6·83-s − 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.917·19-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.152·43-s + 0.875·47-s − 6/7·49-s − 1.23·53-s + 1.56·59-s − 0.128·61-s + 0.855·67-s + 1.78·71-s + 0.468·73-s + 0.341·77-s − 1.12·79-s − 0.658·83-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.59808436449168, −15.11985056734771, −14.57359154588287, −14.15046683046496, −13.60499545084689, −12.85671308001646, −12.54660497861700, −11.78804136092384, −11.36999807412088, −10.87900486565519, −10.19918448224118, −9.564789989380359, −9.141352964567392, −8.374955808676457, −8.071663455031895, −7.154323383162203, −6.717255331628367, −6.161748868722143, −5.390107106204413, −4.699196872100647, −4.170572079755767, −3.543849755979497, −2.542361004505075, −1.987313870026656, −1.099365899600748, 0,
1.099365899600748, 1.987313870026656, 2.542361004505075, 3.543849755979497, 4.170572079755767, 4.699196872100647, 5.390107106204413, 6.161748868722143, 6.717255331628367, 7.154323383162203, 8.071663455031895, 8.374955808676457, 9.141352964567392, 9.564789989380359, 10.19918448224118, 10.87900486565519, 11.36999807412088, 11.78804136092384, 12.54660497861700, 12.85671308001646, 13.60499545084689, 14.15046683046496, 14.57359154588287, 15.11985056734771, 15.59808436449168
