| L(s) = 1 | − 2-s − 3.23·3-s + 4-s + 1.23·5-s + 3.23·6-s − 8-s + 7.47·9-s − 1.23·10-s + 5.23·11-s − 3.23·12-s + 2.47·13-s − 4.00·15-s + 16-s − 17-s − 7.47·18-s + 8.47·19-s + 1.23·20-s − 5.23·22-s + 8·23-s + 3.23·24-s − 3.47·25-s − 2.47·26-s − 14.4·27-s − 5.70·29-s + 4.00·30-s + 1.52·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.86·3-s + 0.5·4-s + 0.552·5-s + 1.32·6-s − 0.353·8-s + 2.49·9-s − 0.390·10-s + 1.57·11-s − 0.934·12-s + 0.685·13-s − 1.03·15-s + 0.250·16-s − 0.242·17-s − 1.76·18-s + 1.94·19-s + 0.276·20-s − 1.11·22-s + 1.66·23-s + 0.660·24-s − 0.694·25-s − 0.484·26-s − 2.78·27-s − 1.05·29-s + 0.730·30-s + 0.274·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9058200887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9058200887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 19 | \( 1 - 8.47T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 1.23T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−9.485562787373402728612197943629, −8.906276169177219611153434760448, −7.36919399626389190051211009433, −6.97408715958172904786754368228, −6.02791788177510173480122476790, −5.65448562109592906960385725339, −4.60733852545660404015810771890, −3.48022857885452924453788531301, −1.57283982348221369821152446271, −0.911195103422984251028119934259,
0.911195103422984251028119934259, 1.57283982348221369821152446271, 3.48022857885452924453788531301, 4.60733852545660404015810771890, 5.65448562109592906960385725339, 6.02791788177510173480122476790, 6.97408715958172904786754368228, 7.36919399626389190051211009433, 8.906276169177219611153434760448, 9.485562787373402728612197943629
