L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + 1.41i·23-s − 1.00i·25-s − 2i·31-s + (0.707 + 0.707i)32-s + (−1.00 + 1.00i)34-s − 1.41i·38-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 + i)19-s + (0.707 + 0.707i)20-s + 1.41i·23-s − 1.00i·25-s − 2i·31-s + (0.707 + 0.707i)32-s + (−1.00 + 1.00i)34-s − 1.41i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7649421831\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7649421831\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + (-1 - i)T + iT^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.16082418165504980075734466990, −9.607022466584688324558734475436, −9.085113752990866780665547868187, −7.963615193229931851982675290965, −7.32067924072005349420657053653, −5.94177936700939655715355141630, −4.96641370577638982848358959965, −3.74322955803488103635970183538, −2.50138346045873189197283043621, −1.24946640468036301038987573849,
1.65089928933564993302833260922, 3.01702673398197661037316773425, 4.69447148435520395844182353373, 5.69949412272248130018087159535, 6.55672195386414276608856355551, 7.13931642202132727536844916255, 8.280599784901348970192876446868, 8.993184408357590596372335303342, 9.927025037500139583340724238653, 10.56172363104129373175838486318