L(s) = 1 | + (0.599 + 0.599i)3-s + (−0.974 + 0.974i)5-s − i·7-s − 2.28i·9-s + (1.72 − 1.72i)11-s + (−1.90 − 1.90i)13-s − 1.16·15-s + 6.71·17-s + (−2.94 − 2.94i)19-s + (0.599 − 0.599i)21-s − 5.29i·23-s + 3.09i·25-s + (3.16 − 3.16i)27-s + (3.03 + 3.03i)29-s + 1.19·31-s + ⋯ |
L(s) = 1 | + (0.346 + 0.346i)3-s + (−0.436 + 0.436i)5-s − 0.377i·7-s − 0.760i·9-s + (0.519 − 0.519i)11-s + (−0.528 − 0.528i)13-s − 0.302·15-s + 1.62·17-s + (−0.676 − 0.676i)19-s + (0.130 − 0.130i)21-s − 1.10i·23-s + 0.619i·25-s + (0.609 − 0.609i)27-s + (0.562 + 0.562i)29-s + 0.215·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52138 - 0.483579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52138 - 0.483579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.599 - 0.599i)T + 3iT^{2} \) |
| 5 | \( 1 + (0.974 - 0.974i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.72 + 1.72i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.90 + 1.90i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 + (2.94 + 2.94i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 + (-3.03 - 3.03i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.94iT - 41T^{2} \) |
| 43 | \( 1 + (-7.02 + 7.02i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + (-3.01 + 3.01i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.96 - 4.96i)T - 59iT^{2} \) |
| 61 | \( 1 + (-9.69 - 9.69i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.55 + 3.55i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 4.06T + 79T^{2} \) |
| 83 | \( 1 + (9.17 + 9.17i)T + 83iT^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−10.11694819999019411617248331402, −9.177601729517991822684703740595, −8.426858763186084115798030400732, −7.48212466426158811077041254112, −6.70151636009945957150588615718, −5.72281458863592554082137619182, −4.47946067279548672184894860436, −3.55584232632482166547017918155, −2.82280997273463673014195018821, −0.818174801406378899699886669253,
1.43842225129493109868235269547, 2.58412696539106938957295299688, 3.91425067544564972436480251410, 4.84240847812902084339112012432, 5.81891016779094809501155094880, 6.93823509282059817238728594438, 7.899752592536011456012591073009, 8.234295801127825386868744269525, 9.424225553744331797629358897877, 9.994027139038798213163014889142