L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s − i·10-s + (−0.366 + 1.36i)13-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)20-s + (−0.866 − 0.499i)25-s − 1.41i·26-s + (−0.707 + 1.22i)29-s + (−0.258 + 0.965i)32-s + (−1 + i)37-s + (−0.5 − 0.866i)40-s + (1.22 − 0.707i)41-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.707 + 0.707i)8-s − i·10-s + (−0.366 + 1.36i)13-s + (0.500 − 0.866i)16-s + (0.258 + 0.965i)20-s + (−0.866 − 0.499i)25-s − 1.41i·26-s + (−0.707 + 1.22i)29-s + (−0.258 + 0.965i)32-s + (−1 + i)37-s + (−0.5 − 0.866i)40-s + (1.22 − 0.707i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5382975431\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5382975431\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.258 - 0.965i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.731340987448029854701496864949, −9.136593175126086093083957831853, −8.274001080976522607982228310549, −7.36206678453126129196598537334, −6.88392466371099360546275962029, −6.18280521804576523509762220351, −5.08446267468287852723430880817, −3.81187737912489637959534621995, −2.71143388325920194190630572401, −1.70232108226614262334799508408,
0.56277024507619513247984376814, 1.92965298235769861204746495724, 3.11911721403236545769765482694, 4.14110564871930074621456468835, 5.31667075219334158662812883099, 6.08591603271269390542203648919, 7.31247195029840901605642585698, 7.903066882469458247115470395351, 8.493429447916081865194537877393, 9.369707911908180682863424241802