L(s) = 1 | + (0.952 − 0.303i)2-s + (0.816 − 0.577i)4-s + (0.881 + 0.473i)5-s + (0.992 − 0.122i)7-s + (0.602 − 0.798i)8-s + (0.982 + 0.183i)10-s + (−0.952 + 0.303i)11-s + (−0.389 + 0.920i)13-s + (0.908 − 0.417i)14-s + (0.332 − 0.943i)16-s + (−0.445 + 0.895i)17-s + (0.932 − 0.361i)19-s + (0.992 − 0.122i)20-s + (−0.816 + 0.577i)22-s + (0.952 + 0.303i)23-s + ⋯ |
L(s) = 1 | + (0.952 − 0.303i)2-s + (0.816 − 0.577i)4-s + (0.881 + 0.473i)5-s + (0.992 − 0.122i)7-s + (0.602 − 0.798i)8-s + (0.982 + 0.183i)10-s + (−0.952 + 0.303i)11-s + (−0.389 + 0.920i)13-s + (0.908 − 0.417i)14-s + (0.332 − 0.943i)16-s + (−0.445 + 0.895i)17-s + (0.932 − 0.361i)19-s + (0.992 − 0.122i)20-s + (−0.816 + 0.577i)22-s + (0.952 + 0.303i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.489197698 - 0.2409227831i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.489197698 - 0.2409227831i\) |
\(L(1)\) |
\(\approx\) |
\(2.264658954 - 0.2070568910i\) |
\(L(1)\) |
\(\approx\) |
\(2.264658954 - 0.2070568910i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (0.952 - 0.303i)T \) |
| 5 | \( 1 + (0.881 + 0.473i)T \) |
| 7 | \( 1 + (0.992 - 0.122i)T \) |
| 11 | \( 1 + (-0.952 + 0.303i)T \) |
| 13 | \( 1 + (-0.389 + 0.920i)T \) |
| 17 | \( 1 + (-0.445 + 0.895i)T \) |
| 19 | \( 1 + (0.932 - 0.361i)T \) |
| 23 | \( 1 + (0.952 + 0.303i)T \) |
| 29 | \( 1 + (0.0307 + 0.999i)T \) |
| 31 | \( 1 + (-0.650 - 0.759i)T \) |
| 37 | \( 1 + (0.273 - 0.961i)T \) |
| 41 | \( 1 + (0.0307 - 0.999i)T \) |
| 43 | \( 1 + (-0.969 - 0.243i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.932 - 0.361i)T \) |
| 59 | \( 1 + (-0.992 - 0.122i)T \) |
| 61 | \( 1 + (0.552 + 0.833i)T \) |
| 67 | \( 1 + (-0.992 - 0.122i)T \) |
| 71 | \( 1 + (-0.850 - 0.526i)T \) |
| 73 | \( 1 + (0.850 + 0.526i)T \) |
| 79 | \( 1 + (-0.0307 - 0.999i)T \) |
| 83 | \( 1 + (-0.992 + 0.122i)T \) |
| 89 | \( 1 + (0.0922 - 0.995i)T \) |
| 97 | \( 1 + (-0.998 - 0.0615i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.76370818899543474220265495166, −21.21445053746723815942199461108, −20.50242090282629408619529356663, −20.045865911893838713746174751418, −18.39270852505203599082268563509, −17.89049754622714263351506918234, −16.99337529673705415022535038431, −16.28206966794869002969240731969, −15.33626742255284414909906357078, −14.69461413131277967803271007310, −13.68044232136911971848864500353, −13.3342467671669254711926683504, −12.39613072938644313894940809176, −11.53889957486052238241166017112, −10.7003365226778424407418977059, −9.77419925297637484297256299577, −8.5062581638079904124088905143, −7.870616158842960815811280464035, −6.91507247760697219879977326509, −5.73206985066169641081723936008, −5.1454551785280482866235775731, −4.67464015143376653700787120978, −3.09433093734768716548712537385, −2.426430080928691059566113278889, −1.25590314625841367278292930584,
1.51516463842736591531485651149, 2.139739597637225181892790889862, 3.09090591656560536339544213439, 4.30695891986975896053430869663, 5.15019060188181469410164653675, 5.74710933612424346980775505236, 6.9688580316450516482607419890, 7.45441149824346694046866509178, 8.92027582249306495632761539154, 9.885684951875077846740508839864, 10.8036788380120433456638745168, 11.20338397058700176571503008981, 12.288238347283028575097289482739, 13.214105353137947085859066664070, 13.76363655318751981884723763742, 14.679308265123994249111447073407, 15.03035193526131174322654080073, 16.169120066334431607266633896762, 17.12398921615889593968622533656, 17.97145446855860815055352916927, 18.680737513616311338732587124998, 19.67627439575532163748491769995, 20.58393162446626565220497780746, 21.22500198836436590498471493264, 21.71983171711480067308141946571