| L(s) = 1 | + 0.284·2-s − 0.918·4-s − 1.30·7-s − 0.546·8-s + 1.91·11-s − 0.372·14-s + 0.763·16-s + 0.830·19-s + 0.546·22-s + 25-s + 1.20·28-s + 1.30·29-s − 0.284·31-s + 0.763·32-s + 0.236·38-s − 1.76·44-s − 0.830·47-s + 0.715·49-s + 0.284·50-s + 0.715·56-s + 0.372·58-s + 1.68·61-s − 0.0810·62-s − 0.546·64-s − 0.763·76-s − 2.51·77-s − 1.04·88-s + ⋯ |
| L(s) = 1 | + 0.284·2-s − 0.918·4-s − 1.30·7-s − 0.546·8-s + 1.91·11-s − 0.372·14-s + 0.763·16-s + 0.830·19-s + 0.546·22-s + 25-s + 1.20·28-s + 1.30·29-s − 0.284·31-s + 0.763·32-s + 0.236·38-s − 1.76·44-s − 0.830·47-s + 0.715·49-s + 0.284·50-s + 0.715·56-s + 0.372·58-s + 1.68·61-s − 0.0810·62-s − 0.546·64-s − 0.763·76-s − 2.51·77-s − 1.04·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9825339280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9825339280\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 2 | \( 1 - 0.284T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.30T + T^{2} \) |
| 11 | \( 1 - 1.91T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.830T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.30T + T^{2} \) |
| 31 | \( 1 + 0.284T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.830T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.68T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.68T + T^{2} \) |
| 97 | \( 1 + 1.91T + T^{2} \) |
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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
−9.598919814410213048834907391565, −9.022189409420759023762561174693, −8.329013050735926897046066730740, −6.95004279159029946225399302471, −6.48414295442907783206870603236, −5.56512512898036632907145867851, −4.51472446103664793662504092443, −3.69732429658509343353195213324, −3.01249136512039291403572252734, −1.07300076749925588822697240767,
1.07300076749925588822697240767, 3.01249136512039291403572252734, 3.69732429658509343353195213324, 4.51472446103664793662504092443, 5.56512512898036632907145867851, 6.48414295442907783206870603236, 6.95004279159029946225399302471, 8.329013050735926897046066730740, 9.022189409420759023762561174693, 9.598919814410213048834907391565
