| L(s) = 1 | − 1.73i·5-s + (0.5 + 2.59i)7-s + 5.19i·11-s − 3.46i·13-s − 3.46i·17-s − 8·19-s − 6.92i·23-s + 2.00·25-s + 6·29-s + 5·31-s + (4.5 − 0.866i)35-s + 4·37-s − 6.92i·41-s + 3.46i·43-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.774i·5-s + (0.188 + 0.981i)7-s + 1.56i·11-s − 0.960i·13-s − 0.840i·17-s − 1.83·19-s − 1.44i·23-s + 0.400·25-s + 1.11·29-s + 0.898·31-s + (0.760 − 0.146i)35-s + 0.657·37-s − 1.08i·41-s + 0.528i·43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.772423858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772423858\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 2.59i)T \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 + 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−8.587022818580158515072550805783, −8.216205831991562722195419660954, −7.14797674551729065439939757771, −6.41970057766840032573831491298, −5.51879362875736544522655995494, −4.67910710826956394426360723021, −4.35741577887307563580969055375, −2.68860226482550610841304715849, −2.20761604454694210534814596204, −0.74172155904591400381856507566,
0.893600883325719009879333510516, 2.14621413959342333315168179823, 3.27849515130741377685159542255, 3.93725464834244646603198141873, 4.74066408934052204389732941642, 6.02015333512657292130556185901, 6.46948651493064809398827198121, 7.13181084832077600128675820844, 8.191060283550992492951725543615, 8.504142080063773725374715809751
