| L(s) = 1 | + 2-s + 4-s − 2·5-s − 3·7-s + 8-s − 2·10-s + 2·11-s + 3·13-s − 3·14-s + 16-s + 17-s − 2·20-s + 2·22-s − 5·23-s − 25-s + 3·26-s − 3·28-s − 3·29-s + 6·31-s + 32-s + 34-s + 6·35-s − 6·37-s − 2·40-s + 12·41-s − 10·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s + 0.353·8-s − 0.632·10-s + 0.603·11-s + 0.832·13-s − 0.801·14-s + 1/4·16-s + 0.242·17-s − 0.447·20-s + 0.426·22-s − 1.04·23-s − 1/5·25-s + 0.588·26-s − 0.566·28-s − 0.557·29-s + 1.07·31-s + 0.176·32-s + 0.171·34-s + 1.01·35-s − 0.986·37-s − 0.316·40-s + 1.87·41-s − 1.52·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.57354574464976765322409383083, −6.82153920123983415441911177877, −6.18887254453205663257211809027, −5.70190777476631346626658577462, −4.55769561592418266967240741583, −3.85687587307837638018027400542, −3.49564033166788137085080968209, −2.58343963510543380433329380507, −1.33932165358990287630226204811, 0,
1.33932165358990287630226204811, 2.58343963510543380433329380507, 3.49564033166788137085080968209, 3.85687587307837638018027400542, 4.55769561592418266967240741583, 5.70190777476631346626658577462, 6.18887254453205663257211809027, 6.82153920123983415441911177877, 7.57354574464976765322409383083
