L(s) = 1 | − 1.73·5-s + (1 + 2.44i)7-s − 4.24i·11-s + 2.44i·13-s − 1.73·17-s + 2.44i·19-s − 8.48i·23-s − 2.00·25-s + 4.24i·29-s − 7.34i·31-s + (−1.73 − 4.24i)35-s + 37-s + 1.73·41-s − 7·43-s + 12.1·47-s + ⋯ |
L(s) = 1 | − 0.774·5-s + (0.377 + 0.925i)7-s − 1.27i·11-s + 0.679i·13-s − 0.420·17-s + 0.561i·19-s − 1.76i·23-s − 0.400·25-s + 0.787i·29-s − 1.31i·31-s + (−0.292 − 0.717i)35-s + 0.164·37-s + 0.270·41-s − 1.06·43-s + 1.76·47-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
\(L(1)\) |
\(\approx\) |
\(1.146748767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.146748767\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 7.34iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 + 2.44iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 2.44iT - 97T^{2} \) |
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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
−8.502472497445711177332229944698, −8.086161472866834679151485858188, −7.09610692152086848787401380155, −6.21208742997116623237146350222, −5.63393415860003010802492175616, −4.58848325592802861323520408796, −3.89797676505784121590230570039, −2.89279373750426527116407326326, −1.95610039643426885534331612010, −0.41978416099709583996406668142,
1.03221639396362627318288334790, 2.21938350226781555797689071923, 3.47333657536795122779936744238, 4.15015694087300773238637750350, 4.86566877578844198242221059250, 5.70893574452764123888537827355, 7.11550294238860296458556176016, 7.16875633179951565472967993363, 8.003232089585694378602081003928, 8.720862094712660525750598068456