| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s + 7·13-s + 2·17-s − 5·19-s − 21-s + 8·23-s + 27-s − 3·29-s − 8·31-s − 33-s − 4·37-s + 7·39-s + 4·41-s + 43-s + 7·47-s − 6·49-s + 2·51-s + 4·53-s − 5·57-s + 12·59-s − 63-s + 10·67-s + 8·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.94·13-s + 0.485·17-s − 1.14·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 0.557·29-s − 1.43·31-s − 0.174·33-s − 0.657·37-s + 1.12·39-s + 0.624·41-s + 0.152·43-s + 1.02·47-s − 6/7·49-s + 0.280·51-s + 0.549·53-s − 0.662·57-s + 1.56·59-s − 0.125·63-s + 1.22·67-s + 0.963·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.081888952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.081888952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.98008159551254, −12.79698548275639, −12.30258335995161, −11.43919753650820, −11.09042492762960, −10.71235411514799, −10.36774174172504, −9.551155078813026, −9.240162688220213, −8.700095740025636, −8.466430156778258, −7.845007613416220, −7.333156376060795, −6.706549338768253, −6.458859986541239, −5.679013650038046, −5.374274683692659, −4.679884150846529, −3.864807576289211, −3.643458843944840, −3.221849416078833, −2.356906749916280, −1.957748241372283, −1.111359457006167, −0.6053561789853568,
0.6053561789853568, 1.111359457006167, 1.957748241372283, 2.356906749916280, 3.221849416078833, 3.643458843944840, 3.864807576289211, 4.679884150846529, 5.374274683692659, 5.679013650038046, 6.458859986541239, 6.706549338768253, 7.333156376060795, 7.845007613416220, 8.466430156778258, 8.700095740025636, 9.240162688220213, 9.551155078813026, 10.36774174172504, 10.71235411514799, 11.09042492762960, 11.43919753650820, 12.30258335995161, 12.79698548275639, 12.98008159551254
