| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s − 7-s + 8-s + 9-s + 3·10-s + 12-s + 13-s − 14-s + 3·15-s + 16-s + 18-s − 2·19-s + 3·20-s − 21-s + 23-s + 24-s + 4·25-s + 26-s + 27-s − 28-s − 9·29-s + 3·30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.670·20-s − 0.218·21-s + 0.208·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.547·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−13.76107689741845, −13.50633570786711, −12.99296775248635, −12.62433190933702, −12.02724647140432, −11.50748470608851, −10.74462597184020, −10.52406782802559, −9.865960767912610, −9.464188656538443, −9.090512077596186, −8.388817989742215, −7.945361142843241, −7.234286144909272, −6.637484442311111, −6.379484439830182, −5.669069784548493, −5.377449124626477, −4.664224318967984, −4.033727476989452, −3.530808460815615, −2.738237137404164, −2.503419553439222, −1.648967911749132, −1.318678762455142, 0,
1.318678762455142, 1.648967911749132, 2.503419553439222, 2.738237137404164, 3.530808460815615, 4.033727476989452, 4.664224318967984, 5.377449124626477, 5.669069784548493, 6.379484439830182, 6.637484442311111, 7.234286144909272, 7.945361142843241, 8.388817989742215, 9.090512077596186, 9.464188656538443, 9.865960767912610, 10.52406782802559, 10.74462597184020, 11.50748470608851, 12.02724647140432, 12.62433190933702, 12.99296775248635, 13.50633570786711, 13.76107689741845
