| L(s) = 1 | + 1.73·5-s + (−1 + 2.44i)7-s − 1.41i·11-s − 2.44i·13-s − 5.19·17-s − 7.34i·19-s + 2.82i·23-s − 2.00·25-s − 7.07i·29-s + 2.44i·31-s + (−1.73 + 4.24i)35-s + 5·37-s − 8.66·41-s − 5·43-s − 8.66·47-s + ⋯ |
| L(s) = 1 | + 0.774·5-s + (−0.377 + 0.925i)7-s − 0.426i·11-s − 0.679i·13-s − 1.26·17-s − 1.68i·19-s + 0.589i·23-s − 0.400·25-s − 1.31i·29-s + 0.439i·31-s + (−0.292 + 0.717i)35-s + 0.821·37-s − 1.35·41-s − 0.762·43-s − 1.26·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.022352505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022352505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 7.34iT - 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 - 2.44iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 8.66T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 - 2.44iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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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
−8.587849190724769268193008647750, −7.889975080912016958346346304804, −6.63821226351572235266400003419, −6.36716699825144413890056922871, −5.37230710686915281675145381744, −4.86604173409420923556668248055, −3.55708490797448211037227225228, −2.64485509824527501329535462750, −1.95575135540242145309739374218, −0.29673155227346828015419320557,
1.42506369053523215070259667415, 2.22086211762820082792330300084, 3.48785239393292837442239817350, 4.24918047807647133560683929828, 5.04955768075865872459875128840, 6.13996899913554701368300562726, 6.60589848657121452755309574855, 7.34399508069847759334977751317, 8.240405561723184330601229118515, 9.020628125646532001553119287822
