| L(s) = 1 | + 11-s + 4·13-s + 8·19-s + 4·23-s + 2·29-s + 4·31-s + 8·37-s + 2·41-s − 4·47-s − 7·49-s − 12·53-s − 12·59-s − 2·61-s + 12·67-s + 12·73-s + 12·79-s + 8·83-s − 6·89-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 1.10·13-s + 1.83·19-s + 0.834·23-s + 0.371·29-s + 0.718·31-s + 1.31·37-s + 0.312·41-s − 0.583·47-s − 49-s − 1.64·53-s − 1.56·59-s − 0.256·61-s + 1.46·67-s + 1.40·73-s + 1.35·79-s + 0.878·83-s − 0.635·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.757355349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.757355349\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.25695968782861, −12.90986765652249, −12.28879344699998, −11.86081154277673, −11.29439337047961, −11.01988059558953, −10.56828115805992, −9.724454752410577, −9.437342867652717, −9.196957398515602, −8.293196514476135, −7.995849746538586, −7.619672905690233, −6.700256563389198, −6.569484934365364, −5.939028886076755, −5.328284536212630, −4.850548414384526, −4.314009910752258, −3.554259416232208, −3.180499640184723, −2.669427101327285, −1.705649406386794, −1.158700967519838, −0.6443717702193690,
0.6443717702193690, 1.158700967519838, 1.705649406386794, 2.669427101327285, 3.180499640184723, 3.554259416232208, 4.314009910752258, 4.850548414384526, 5.328284536212630, 5.939028886076755, 6.569484934365364, 6.700256563389198, 7.619672905690233, 7.995849746538586, 8.293196514476135, 9.196957398515602, 9.437342867652717, 9.724454752410577, 10.56828115805992, 11.01988059558953, 11.29439337047961, 11.86081154277673, 12.28879344699998, 12.90986765652249, 13.25695968782861
