| L(s) = 1 | − 5-s + 7-s − 2·11-s − 5·13-s − 3·17-s + 2·19-s + 6·23-s − 4·25-s + 5·29-s + 6·31-s − 35-s − 3·37-s + 10·41-s + 4·43-s − 6·47-s + 49-s + 6·53-s + 2·55-s + 6·59-s + 7·61-s + 5·65-s + 2·67-s − 12·71-s − 15·73-s − 2·77-s − 14·79-s + 18·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.38·13-s − 0.727·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.928·29-s + 1.07·31-s − 0.169·35-s − 0.493·37-s + 1.56·41-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.269·55-s + 0.781·59-s + 0.896·61-s + 0.620·65-s + 0.244·67-s − 1.42·71-s − 1.75·73-s − 0.227·77-s − 1.57·79-s + 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−7.33746188495184275277307645365, −6.95535832045093033479507349258, −5.97344059812673449221041610599, −5.16344171548902730815717774948, −4.66550004163471097500071123253, −3.98432846150849196605879898918, −2.81040907337827776490959146407, −2.44561759450789663777294638154, −1.14181869646090517252568791694, 0,
1.14181869646090517252568791694, 2.44561759450789663777294638154, 2.81040907337827776490959146407, 3.98432846150849196605879898918, 4.66550004163471097500071123253, 5.16344171548902730815717774948, 5.97344059812673449221041610599, 6.95535832045093033479507349258, 7.33746188495184275277307645365
