L(s) = 1 | + 2i·3-s − 2i·5-s + 4·7-s − 9-s + 2i·11-s + 2i·13-s + 4·15-s − 2·17-s + 2i·19-s + 8i·21-s − 4·23-s + 25-s + 4i·27-s − 6i·29-s − 4·33-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 0.894i·5-s + 1.51·7-s − 0.333·9-s + 0.603i·11-s + 0.554i·13-s + 1.03·15-s − 0.485·17-s + 0.458i·19-s + 1.74i·21-s − 0.834·23-s + 0.200·25-s + 0.769i·27-s − 1.11i·29-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31847 + 0.546129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31847 + 0.546129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 14iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.95229495174916424715383421785, −11.14494699170078649725293233429, −10.19846153151799484232110380476, −9.256108972438302022834978407199, −8.468954768123919982906953682469, −7.42675807818385756312028625007, −5.65174324111818920691717656115, −4.59266733615629306970623763094, −4.18750880843649192496613275121, −1.84807485602538521290021290184,
1.48527687479949932962438323565, 2.86673688421774418436559376471, 4.67760979703143973951450660168, 6.05194408029281576944524179606, 7.03622164116349230603131877481, 7.86528301643183403367527227627, 8.611783944663925494202640188699, 10.26577727041265197797122910912, 11.15648399208933492725911632942, 11.75762837471476313294134706289