L(s) = 1 | + (1.16 + 0.795i)2-s + (0.735 + 1.85i)4-s − 2.09·5-s − i·7-s + (−0.618 + 2.75i)8-s + (−2.44 − 1.66i)10-s − 0.961i·11-s + 4.60i·13-s + (0.795 − 1.16i)14-s + (−2.91 + 2.73i)16-s + 3.28i·17-s + 3.66·19-s + (−1.53 − 3.89i)20-s + (0.764 − 1.12i)22-s − 2.40·23-s + ⋯ |
L(s) = 1 | + (0.826 + 0.562i)2-s + (0.367 + 0.929i)4-s − 0.936·5-s − 0.377i·7-s + (−0.218 + 0.975i)8-s + (−0.774 − 0.526i)10-s − 0.289i·11-s + 1.27i·13-s + (0.212 − 0.312i)14-s + (−0.729 + 0.684i)16-s + 0.795i·17-s + 0.841·19-s + (−0.344 − 0.870i)20-s + (0.162 − 0.239i)22-s − 0.501·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.409833901\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409833901\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 - 0.795i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.09T + 5T^{2} \) |
| 11 | \( 1 + 0.961iT - 11T^{2} \) |
| 13 | \( 1 - 4.60iT - 13T^{2} \) |
| 17 | \( 1 - 3.28iT - 17T^{2} \) |
| 19 | \( 1 - 3.66T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 9.31T + 29T^{2} \) |
| 31 | \( 1 - 1.84iT - 31T^{2} \) |
| 37 | \( 1 - 2.29iT - 37T^{2} \) |
| 41 | \( 1 + 0.314iT - 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 + 2.34T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 - 7.99iT - 59T^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 1.26iT - 79T^{2} \) |
| 83 | \( 1 + 2.99iT - 83T^{2} \) |
| 89 | \( 1 + 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 4.98T + 97T^{2} \) |
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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
−9.752063313888223342113890749344, −8.784196319990698871004004021492, −8.003608745494226056561039907935, −7.37570041117983779527760137763, −6.63888998098335324145463370301, −5.76144423667087570173632552082, −4.73979288824627005650425291263, −3.91814903658145108012064555279, −3.39169482534188378731767632522, −1.85603307638737927082819212337,
0.40002091001062807850041873921, 2.00688972714996321857255271776, 3.20683320690640012286943984531, 3.76411485555016258164095784416, 4.96609506644748118573705800558, 5.47799238725662793602541842064, 6.54203755790428005467357384349, 7.54691055567353333805945901933, 8.091402867678996726674247623263, 9.451586296483445461286785416392