L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.41 + 2.44i)5-s − 0.999·8-s + (1.5 − 2.59i)9-s + (1.41 + 2.44i)10-s + (−0.5 − 0.866i)11-s − 5.65·13-s + (−0.5 + 0.866i)16-s + (1.41 + 2.44i)17-s + (−1.5 − 2.59i)18-s + (4.24 − 7.34i)19-s + 2.82·20-s − 0.999·22-s + (4 − 6.92i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.632 + 1.09i)5-s − 0.353·8-s + (0.5 − 0.866i)9-s + (0.447 + 0.774i)10-s + (−0.150 − 0.261i)11-s − 1.56·13-s + (−0.125 + 0.216i)16-s + (0.342 + 0.594i)17-s + (−0.353 − 0.612i)18-s + (0.973 − 1.68i)19-s + 0.632·20-s − 0.213·22-s + (0.834 − 1.44i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.163836721\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.163836721\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + (-1.41 - 2.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.24 + 7.34i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (4.24 + 7.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 - 4.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (4.24 + 7.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + (-5.65 + 9.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−9.616883414583547896249114625400, −9.139171584939684717806923604655, −7.63057660440348854854983408918, −7.16120151184465327456370574991, −6.25848922324922639371056781300, −5.06729461168235260334755207169, −4.12232062349114558505441557990, −3.17192163934378238436428927005, −2.41418343712792240296525227159, −0.47091112443030590078212435359,
1.52711398307412433780369009222, 3.17528496827310334658313984916, 4.32397177036322888782790060971, 5.11379231616085763030492813835, 5.51635620883250021461696594642, 7.25431588565736251856680769295, 7.52056555745825945939714066809, 8.228592552766883094168777485754, 9.413367222499596184376416519181, 9.829830415618208863815785975058