L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.5 + 0.133i)7-s + (0.707 − 0.707i)8-s + (−0.500 − 0.866i)10-s + (−1.67 + 0.965i)11-s + (−0.258 + 0.448i)14-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)20-s + (−0.500 − 1.86i)22-s − 1.00i·25-s + (−0.366 − 0.366i)28-s + (−0.707 − 1.22i)29-s + (−0.866 + 1.5i)31-s + (−0.965 + 0.258i)32-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (0.5 + 0.133i)7-s + (0.707 − 0.707i)8-s + (−0.500 − 0.866i)10-s + (−1.67 + 0.965i)11-s + (−0.258 + 0.448i)14-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)20-s + (−0.500 − 1.86i)22-s − 1.00i·25-s + (−0.366 − 0.366i)28-s + (−0.707 − 1.22i)29-s + (−0.866 + 1.5i)31-s + (−0.965 + 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1816101113\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1816101113\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 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 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.235563041948502953998594111169, −8.205772527094896920680387283333, −7.84028393955969408628780374929, −7.26664131912722009875622222963, −6.55672505521014256234076913583, −5.54245480590726911859013721876, −4.91149421241052669710881529426, −4.16257458583418148355440108080, −3.04763506723218620952702420036, −1.89046195208382945291470499492,
0.12057715380291645348342933576, 1.43714985078773843832786085138, 2.61359381801972149971244150996, 3.48572460452837638255854618727, 4.28579029775036856130457267670, 5.18370784242479731739175199954, 5.59822776074310600873981021312, 7.22463248633985414664799208955, 7.954596226640945538084139978653, 8.266657557004188996659509372886