L(s) = 1 | + (0.765 − 0.765i)3-s + (−0.414 − 0.414i)5-s + 3.69i·7-s + 1.82i·9-s + (2.93 + 2.93i)11-s + (−2.41 + 2.41i)13-s − 0.634·15-s − 2.82·17-s + (4.46 − 4.46i)19-s + (2.82 + 2.82i)21-s + 6.75i·23-s − 4.65i·25-s + (3.69 + 3.69i)27-s + (5.24 − 5.24i)29-s + 3.06·31-s + ⋯ |
L(s) = 1 | + (0.441 − 0.441i)3-s + (−0.185 − 0.185i)5-s + 1.39i·7-s + 0.609i·9-s + (0.883 + 0.883i)11-s + (−0.669 + 0.669i)13-s − 0.163·15-s − 0.685·17-s + (1.02 − 1.02i)19-s + (0.617 + 0.617i)21-s + 1.40i·23-s − 0.931i·25-s + (0.711 + 0.711i)27-s + (0.973 − 0.973i)29-s + 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44376 + 0.598027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44376 + 0.598027i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.765 + 0.765i)T - 3iT^{2} \) |
| 5 | \( 1 + (0.414 + 0.414i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.69iT - 7T^{2} \) |
| 11 | \( 1 + (-2.93 - 2.93i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.41 - 2.41i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.75iT - 23T^{2} \) |
| 29 | \( 1 + (-5.24 + 5.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + (6.41 + 6.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + (0.765 + 0.765i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + (-3.24 - 3.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.765 + 0.765i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.757 - 0.757i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.39 + 1.39i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.02iT - 71T^{2} \) |
| 73 | \( 1 - 6.48iT - 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + (9.68 - 9.68i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.82iT - 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−11.30046220237527290527274160891, −9.807007087823407148498164292308, −9.187726588167985210274003849003, −8.406730688118521605584328630026, −7.39221152772756911989995743295, −6.59321124266476697231590617746, −5.28816517649647301351178938471, −4.42523014947871481965286996158, −2.73269470059562279233877530719, −1.87131517503645077205991377619,
0.952473800402299021182891169430, 3.16278139991725190232395647040, 3.78897788530362012449288083232, 4.88620447094572529493857703131, 6.38299514463472249913779275276, 7.10032334470527661461795873409, 8.193839155548104857616788427737, 8.999508783826347439542720567412, 10.07884835419968405389257402228, 10.52061854213044698454359106857