| L(s) = 1 | − 2·3-s + 9-s + 3·11-s − 4·13-s − 2·19-s + 3·23-s + 4·27-s + 9·29-s − 8·31-s − 6·33-s − 5·37-s + 8·39-s + 6·41-s − 11·43-s + 6·47-s − 6·53-s + 4·57-s + 10·61-s − 5·67-s − 6·69-s + 15·71-s − 10·73-s − 7·79-s − 11·81-s + 12·83-s − 18·87-s + 12·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 0.458·19-s + 0.625·23-s + 0.769·27-s + 1.67·29-s − 1.43·31-s − 1.04·33-s − 0.821·37-s + 1.28·39-s + 0.937·41-s − 1.67·43-s + 0.875·47-s − 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.610·67-s − 0.722·69-s + 1.78·71-s − 1.17·73-s − 0.787·79-s − 1.22·81-s + 1.31·83-s − 1.92·87-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.80589246287640264791392714277, −6.80701054258148316233147945395, −6.63981507210352316787384177315, −5.64540591646809451157330206410, −5.04099022743854910088202526014, −4.39593268428752388877791405137, −3.38164593300486549819108720948, −2.32345747823811902504887844044, −1.14273855925415900380919653773, 0,
1.14273855925415900380919653773, 2.32345747823811902504887844044, 3.38164593300486549819108720948, 4.39593268428752388877791405137, 5.04099022743854910088202526014, 5.64540591646809451157330206410, 6.63981507210352316787384177315, 6.80701054258148316233147945395, 7.80589246287640264791392714277
