L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.133 + 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.499 + 0.866i)10-s + (1.67 − 0.965i)11-s + (−0.258 + 0.448i)14-s + (0.500 + 0.866i)16-s + (−0.707 + 0.707i)20-s + (1.86 − 0.500i)22-s + (−0.866 − 0.499i)25-s + (−0.366 + 0.366i)28-s + (−0.707 − 1.22i)29-s + (−0.866 + 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.258 + 0.965i)5-s + (−0.133 + 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.499 + 0.866i)10-s + (1.67 − 0.965i)11-s + (−0.258 + 0.448i)14-s + (0.500 + 0.866i)16-s + (−0.707 + 0.707i)20-s + (1.86 − 0.500i)22-s + (−0.866 − 0.499i)25-s + (−0.366 + 0.366i)28-s + (−0.707 − 1.22i)29-s + (−0.866 + 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.396973712\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.396973712\) |
\(L(1)\) |
|
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 | \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−8.834025022185097835429641022533, −8.119138459486429171399333638855, −7.20299343657055076316903050864, −6.58099720047923832632459713777, −6.07367859925468404049019244202, −5.31146451246044906713933253188, −4.09223264876670743477452902506, −3.58483005045698098383672754032, −2.82361972808960022056647935473, −1.71785411436816197446835380655,
1.20268342033917919403635596394, 1.97766213417808607746720641575, 3.44183915598758441969473026664, 4.12703806495809847297497593397, 4.57864670087177356376545857760, 5.54599361013851687590599123344, 6.29600297107323960331302158391, 7.20174741064717009572609679790, 7.59804416566065847123059535542, 8.933504972919320233567861842991