L(s) = 1 | + (−0.225 − 1.39i)2-s + (−0.687 − 1.58i)3-s + (−1.89 + 0.629i)4-s + 2.07i·5-s + (−2.06 + 1.31i)6-s + (1.30 + 2.50i)8-s + (−2.05 + 2.18i)9-s + (2.89 − 0.467i)10-s + 3.98·11-s + (2.30 + 2.58i)12-s − 1.30·13-s + (3.30 − 1.42i)15-s + (3.20 − 2.38i)16-s + 2.94i·17-s + (3.51 + 2.37i)18-s + 1.09i·19-s + ⋯ |
L(s) = 1 | + (−0.159 − 0.987i)2-s + (−0.397 − 0.917i)3-s + (−0.949 + 0.314i)4-s + 0.928i·5-s + (−0.842 + 0.538i)6-s + (0.461 + 0.887i)8-s + (−0.684 + 0.729i)9-s + (0.916 − 0.147i)10-s + 1.20·11-s + (0.665 + 0.746i)12-s − 0.363·13-s + (0.852 − 0.368i)15-s + (0.802 − 0.597i)16-s + 0.713i·17-s + (0.828 + 0.559i)18-s + 0.251i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.972109 - 0.435498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.972109 - 0.435498i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.225 + 1.39i)T \) |
| 3 | \( 1 + (0.687 + 1.58i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.07iT - 5T^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 2.94iT - 17T^{2} \) |
| 19 | \( 1 - 1.09iT - 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 - 0.865iT - 29T^{2} \) |
| 31 | \( 1 + 3.68iT - 31T^{2} \) |
| 37 | \( 1 - 4.17T + 37T^{2} \) |
| 41 | \( 1 - 7.01iT - 41T^{2} \) |
| 43 | \( 1 + 4.27iT - 43T^{2} \) |
| 47 | \( 1 + 7.50T + 47T^{2} \) |
| 53 | \( 1 - 4.94iT - 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 - 4.10T + 61T^{2} \) |
| 67 | \( 1 - 2.42iT - 67T^{2} \) |
| 71 | \( 1 - 0.901T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 8.52iT - 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−10.94377792552211002501542991404, −9.875425604906339526473914587470, −8.929329480652154585840546466479, −7.945469326323192282675540905240, −6.98379894272566708456571117791, −6.18020916042285844931476213633, −4.88462578870067729342467284260, −3.54450868714336197584207174281, −2.47639584469073646947280027574, −1.20956593449698827988826975542,
0.846182178130867816662385605717, 3.50431943822292433862268224271, 4.72299210770913969492729710348, 5.04909198156483343486377746835, 6.25691744217177068415543468772, 7.08622384022998982566937843446, 8.387988490136078418262195395455, 9.234124454814871804082830837264, 9.428836823010178233890073043573, 10.63878072235343081386836454577