| L(s) = 1 | − 3.46i·5-s + (2 − 1.73i)7-s − 3.46i·11-s − 3.46i·13-s + 2·19-s − 3.46i·23-s − 6.99·25-s + 6·29-s + 8·31-s + (−5.99 − 6.92i)35-s + 2·37-s + 6.92i·41-s − 10.3i·43-s + (1.00 − 6.92i)49-s + 6·53-s + ⋯ |
| L(s) = 1 | − 1.54i·5-s + (0.755 − 0.654i)7-s − 1.04i·11-s − 0.960i·13-s + 0.458·19-s − 0.722i·23-s − 1.39·25-s + 1.11·29-s + 1.43·31-s + (−1.01 − 1.17i)35-s + 0.328·37-s + 1.08i·41-s − 1.58i·43-s + (0.142 − 0.989i)49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.144603960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.144603960\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.311492199513727407679115050624, −7.74632956708739299467140651858, −6.64929427669292913537578403352, −5.75640528600867934707596118030, −5.06081422891627967285505970636, −4.53999919407921005440800133066, −3.65368054408025903311939283254, −2.56074067948233261186936320156, −1.10721803633979057332405363837, −0.73005477831156902975680407883,
1.56681753392349750065218013904, 2.42795451632466793796997244096, 3.09548094053656136289574167579, 4.24952697708590079815601943601, 4.87871780078869952405542984790, 5.95299128062302412355606174340, 6.57071922797863295308727580018, 7.26804388829429283681088163736, 7.80706935359832165938737593226, 8.704333491648407150142376325945
