| L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·13-s − 6·17-s − 2·19-s + 2·21-s − 3·23-s − 4·27-s + 6·29-s + 7·31-s − 4·37-s + 8·39-s − 9·41-s − 4·43-s + 9·47-s − 6·49-s − 12·51-s − 4·57-s + 6·59-s − 10·61-s + 63-s + 10·67-s − 6·69-s + 11·73-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 0.436·21-s − 0.625·23-s − 0.769·27-s + 1.11·29-s + 1.25·31-s − 0.657·37-s + 1.28·39-s − 1.40·41-s − 0.609·43-s + 1.31·47-s − 6/7·49-s − 1.68·51-s − 0.529·57-s + 0.781·59-s − 1.28·61-s + 0.125·63-s + 1.22·67-s − 0.722·69-s + 1.28·73-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.58900452135851, −13.09602306511559, −12.26571860282502, −12.09685509246297, −11.29842407295075, −11.07222307609668, −10.40460799510374, −10.10628867783152, −9.270820785458116, −9.079917159797013, −8.422703191938192, −8.159825123083247, −8.002018841611112, −6.967531519918451, −6.640492167772100, −6.239644624324071, −5.517763730899923, −4.841083177854394, −4.416343313566330, −3.757470228970751, −3.431129718466217, −2.669803373265104, −2.215940125255978, −1.715188025762015, −0.9417947421822164, 0,
0.9417947421822164, 1.715188025762015, 2.215940125255978, 2.669803373265104, 3.431129718466217, 3.757470228970751, 4.416343313566330, 4.841083177854394, 5.517763730899923, 6.239644624324071, 6.640492167772100, 6.967531519918451, 8.002018841611112, 8.159825123083247, 8.422703191938192, 9.079917159797013, 9.270820785458116, 10.10628867783152, 10.40460799510374, 11.07222307609668, 11.29842407295075, 12.09685509246297, 12.26571860282502, 13.09602306511559, 13.58900452135851
