| L(s) = 1 | − i·2-s − 4-s + 0.125·5-s + 3.10·7-s + i·8-s − 0.125i·10-s + 4.94·11-s − 4.70i·13-s − 3.10i·14-s + 16-s − 2.61i·17-s − 2.28·19-s − 0.125·20-s − 4.94i·22-s − 5.47·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.0560·5-s + 1.17·7-s + 0.353i·8-s − 0.0396i·10-s + 1.49·11-s − 1.30i·13-s − 0.829i·14-s + 0.250·16-s − 0.633i·17-s − 0.523·19-s − 0.0280·20-s − 1.05i·22-s − 1.14·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.886586148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886586148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 223 | \( 1 + (-14.6 + 2.91i)T \) |
| good | 5 | \( 1 - 0.125T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 + 4.70iT - 13T^{2} \) |
| 17 | \( 1 + 2.61iT - 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 2.70iT - 29T^{2} \) |
| 31 | \( 1 + 0.559T + 31T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 11.5iT - 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 2.06iT - 47T^{2} \) |
| 53 | \( 1 + 7.30iT - 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 1.60iT - 61T^{2} \) |
| 67 | \( 1 - 1.97iT - 67T^{2} \) |
| 71 | \( 1 + 9.30T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 17.2iT - 79T^{2} \) |
| 83 | \( 1 - 3.82iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 3.12iT - 97T^{2} \) |
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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
−8.311764059365666288325382599709, −7.63637899970457955386641269593, −6.73724620848076202581455643615, −5.70737503901647433347726759180, −5.18172406751776211062702940985, −4.09090245920340776546583231725, −3.71693581494577224022995168835, −2.38820272842234663777159393448, −1.67612194223541806291314649802, −0.55250807488804041629599764622,
1.39397910099971895198870470584, 2.00943034491713118887064510038, 3.72822122741427076425489752077, 4.23831982357878782093563502134, 4.90452251367469531502803516307, 6.01401054214857822397920673093, 6.42228324681374989024480536018, 7.23976511352384044062574324833, 8.014285335409374234806551455936, 8.620044862636775174688009866502
