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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.698479425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.698479425\) |
\(L(\frac{3}{2})\) |
|
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
−9.154480413383507872776540143350, −8.230659355500302233329943368466, −7.44158493876838305652500989366, −6.64736846153915600103018875532, −6.23791272768971002080021076306, −5.05005180770035736117198639534, −4.47239690492159970291767306484, −3.26290870978848290754273136398, −2.50456312538850418832409855782, −1.58946466492516623393880001500,
0.60253261047836479686556714910, 1.20458629331792589701431308306, 3.09037940669844404205461566467, 3.41458956676008800958214186861, 4.62065043187181382590848178771, 5.43383286617479293384194166205, 5.90972994336019980328210711859, 7.22293875315507179668820537662, 7.56147787022240455212107554339, 8.485748344008287363452887878158