L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s + 5·13-s + 16-s + 6·17-s − 4·19-s − 6·22-s + 6·23-s − 5·25-s − 5·26-s + 6·29-s − 31-s − 32-s − 6·34-s − 37-s + 4·38-s − 6·41-s − 43-s + 6·44-s − 6·46-s − 6·47-s + 5·50-s + 5·52-s − 6·53-s − 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s + 1.38·13-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 1.27·22-s + 1.25·23-s − 25-s − 0.980·26-s + 1.11·29-s − 0.179·31-s − 0.176·32-s − 1.02·34-s − 0.164·37-s + 0.648·38-s − 0.937·41-s − 0.152·43-s + 0.904·44-s − 0.884·46-s − 0.875·47-s + 0.707·50-s + 0.693·52-s − 0.824·53-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.668142139\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.668142139\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.807440564139870105204971769637, −8.314641853250403750862892328977, −7.41884840066332128974372791778, −6.38673320168995007470906447914, −6.25230709459219220953428236513, −4.97773503051827088090744606249, −3.81520318696156667430997258337, −3.25978066607203899213920217212, −1.72172496964637188339495477799, −0.999334210364589934822853983965,
0.999334210364589934822853983965, 1.72172496964637188339495477799, 3.25978066607203899213920217212, 3.81520318696156667430997258337, 4.97773503051827088090744606249, 6.25230709459219220953428236513, 6.38673320168995007470906447914, 7.41884840066332128974372791778, 8.314641853250403750862892328977, 8.807440564139870105204971769637