L(s) = 1 | − 2.41i·2-s − 0.537i·3-s − 3.85·4-s + (−2.07 + 0.826i)5-s − 1.29·6-s − 3.18i·7-s + 4.49i·8-s + 2.71·9-s + (2 + 5.02i)10-s + 4.15·11-s + 2.07i·12-s − 2.07i·13-s − 7.71·14-s + (0.443 + 1.11i)15-s + 3.15·16-s + 5.79i·17-s + ⋯ |
L(s) = 1 | − 1.71i·2-s − 0.310i·3-s − 1.92·4-s + (−0.929 + 0.369i)5-s − 0.530·6-s − 1.20i·7-s + 1.58i·8-s + 0.903·9-s + (0.632 + 1.58i)10-s + 1.25·11-s + 0.597i·12-s − 0.574i·13-s − 2.06·14-s + (0.114 + 0.288i)15-s + 0.788·16-s + 1.40i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.161267 - 0.841718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.161267 - 0.841718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.07 - 0.826i)T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 2.41iT - 2T^{2} \) |
| 3 | \( 1 + 0.537iT - 3T^{2} \) |
| 7 | \( 1 + 3.18iT - 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 + 2.07iT - 13T^{2} \) |
| 17 | \( 1 - 5.79iT - 17T^{2} \) |
| 23 | \( 1 - 2.60iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 - 4.30iT - 37T^{2} \) |
| 41 | \( 1 + 0.599T + 41T^{2} \) |
| 43 | \( 1 + 3.18iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 - 4.76iT - 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 2.72iT - 73T^{2} \) |
| 79 | \( 1 - 1.40T + 79T^{2} \) |
| 83 | \( 1 - 7.07iT - 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 2.07iT - 97T^{2} \) |
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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
−13.12467186518502437031035265611, −12.36096432927352159860279984005, −11.34510190046378379044315038438, −10.56916203529522913710147082227, −9.653356311466335909467732732958, −8.078641367623309618278074020147, −6.84997531671277012796673575455, −4.22537240593369455802252818368, −3.60313810374584796487520343174, −1.29396656358061951707671114762,
4.13013001068348509602005960455, 5.18809842675337315714788227610, 6.63054945188045570305069337188, 7.56623712770699265231123577404, 8.915038587374742941916063910158, 9.354185330099607118717815923817, 11.53950145719745279541421919243, 12.44442891085560564719219286913, 13.84451991241822544180007603552, 14.91587755828743996140732734371