L(s) = 1 | + (0.781 − 1.54i)3-s + 1.94i·5-s − 3.97i·7-s + (−1.77 − 2.41i)9-s + 5.57·11-s − 1.25·13-s + (3.00 + 1.51i)15-s − i·17-s + 2.07i·19-s + (−6.13 − 3.10i)21-s − 4.52·23-s + 1.22·25-s + (−5.12 + 0.861i)27-s − 7.39i·29-s − 8.94i·31-s + ⋯ |
L(s) = 1 | + (0.451 − 0.892i)3-s + 0.868i·5-s − 1.50i·7-s + (−0.592 − 0.805i)9-s + 1.68·11-s − 0.347·13-s + (0.775 + 0.391i)15-s − 0.242i·17-s + 0.476i·19-s + (−1.33 − 0.677i)21-s − 0.942·23-s + 0.245·25-s + (−0.986 + 0.165i)27-s − 1.37i·29-s − 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0554 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0554 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29905 - 1.22885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29905 - 1.22885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.781 + 1.54i)T \) |
| 17 | \( 1 + iT \) |
good | 5 | \( 1 - 1.94iT - 5T^{2} \) |
| 7 | \( 1 + 3.97iT - 7T^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 19 | \( 1 - 2.07iT - 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 7.39iT - 29T^{2} \) |
| 31 | \( 1 + 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 - 0.747iT - 41T^{2} \) |
| 43 | \( 1 - 4.26iT - 43T^{2} \) |
| 47 | \( 1 - 4.99T + 47T^{2} \) |
| 53 | \( 1 - 1.80iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 0.151T + 71T^{2} \) |
| 73 | \( 1 + 1.15T + 73T^{2} \) |
| 79 | \( 1 - 3.97iT - 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 17.9iT - 89T^{2} \) |
| 97 | \( 1 + 9.97T + 97T^{2} \) |
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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
−9.923828424491718215593788368349, −9.318337071109692961183939076760, −7.988988707831088290867209477974, −7.47260858727076606209202654345, −6.62070764229462451730823643204, −6.14655507460096656353607520827, −4.20480532594571398584201504919, −3.60152252751026516785042244437, −2.25217443930298078680976128112, −0.898261902428663063453086382158,
1.73173625797538186482161559107, 3.03154263506468326164625746122, 4.15226341802177869854611601813, 5.05256033063839949379854368854, 5.79447800881839376449178930399, 6.96302762956142980362045801646, 8.466919085951513100379367393908, 8.834582083361919032736443884772, 9.290407731976596293758642744831, 10.24631291744475008638404210363