L(s) = 1 | + i·3-s − 9-s + (1 − i)11-s + 2i·17-s + (1 − i)19-s + i·25-s − i·27-s + (1 + i)33-s + (1 + i)43-s + 49-s − 2·51-s + (1 + i)57-s + (−1 + i)59-s + (1 − i)67-s − 75-s + ⋯ |
L(s) = 1 | + i·3-s − 9-s + (1 − i)11-s + 2i·17-s + (1 − i)19-s + i·25-s − i·27-s + (1 + i)33-s + (1 + i)43-s + 49-s − 2·51-s + (1 + i)57-s + (−1 + i)59-s + (1 − i)67-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.287371027\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.287371027\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (1 - i)T - iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1 - i)T + iT^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 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
−9.120861765805167954778907475914, −8.480720198105445650540411152298, −7.69609808001593294661404565500, −6.56792198876660692623188299498, −5.91338130275833326755794402507, −5.23607029821142325538584616136, −4.16425258856899897415220870293, −3.64573613087210841047816004284, −2.78031566266009699371606239922, −1.27070070165562505454868999107,
0.943485650547084277800167327003, 2.04257829060276103367834255572, 2.92104513493754761761269315478, 4.00970815369222455794927578997, 5.01203155720588031419351183405, 5.78976522458216297396938701614, 6.69901979833410237389365337597, 7.23648074327990459567691904638, 7.77401081529444014647453021829, 8.742067011689231100751771517544