| L(s) = 1 | + 1.61i·3-s + 0.618i·7-s + 0.381·9-s + 4.63·11-s − 4.63i·13-s + 7.50i·17-s − 7.50·19-s − 1.00·21-s − 4.61i·23-s + 5.47i·27-s + 10.0·29-s − 4.63·31-s + 7.50i·33-s − 2.86i·37-s + 7.50·39-s + ⋯ |
| L(s) = 1 | + 0.934i·3-s + 0.233i·7-s + 0.127·9-s + 1.39·11-s − 1.28i·13-s + 1.82i·17-s − 1.72·19-s − 0.218·21-s − 0.962i·23-s + 1.05i·27-s + 1.87·29-s − 0.833·31-s + 1.30i·33-s − 0.471i·37-s + 1.20·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045482961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045482961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.61iT - 3T^{2} \) |
| 7 | \( 1 - 0.618iT - 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 + 4.63iT - 13T^{2} \) |
| 17 | \( 1 - 7.50iT - 17T^{2} \) |
| 19 | \( 1 + 7.50T + 19T^{2} \) |
| 23 | \( 1 + 4.61iT - 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + 2.86iT - 37T^{2} \) |
| 41 | \( 1 + 0.854T + 41T^{2} \) |
| 43 | \( 1 - 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 9.27iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 1.52iT - 67T^{2} \) |
| 71 | \( 1 - 7.50T + 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 - 7.56iT - 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−8.510479783363709412132053863036, −8.318029366587653047888160432561, −7.04495768900563695729393784423, −6.28168208173730776326092101640, −5.77419171206552223966325471386, −4.57491401038162028604853061324, −4.16528987718960114160712951621, −3.43182410906491614551828974976, −2.29044976993426780892005412851, −1.10157069213622129244525142039,
0.68297509971162117950912920572, 1.69201047273722483810939653880, 2.42100394661581418740121015500, 3.78454744475473825904213194454, 4.34745373760192315377419290788, 5.29267744534205828491445866847, 6.51022527462073768743624980098, 6.82463952348028599811133482830, 7.18392081619419449209245729257, 8.341484176209007022373468854918
