| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 5·13-s − 14-s + 16-s + 6·17-s + 5·19-s − 20-s − 3·23-s + 25-s + 5·26-s − 28-s + 8·31-s + 32-s + 6·34-s + 35-s + 2·37-s + 5·38-s − 40-s − 3·41-s − 4·43-s − 3·46-s − 9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s − 0.223·20-s − 0.625·23-s + 1/5·25-s + 0.980·26-s − 0.188·28-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s + 0.811·38-s − 0.158·40-s − 0.468·41-s − 0.609·43-s − 0.442·46-s − 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.345492780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.345492780\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−10.30328332414120198497263720819, −9.566831202589947576418410084981, −8.312293789871667127278945468914, −7.71078313759863912854007134813, −6.57624001584826522381764449343, −5.85900984964464586455726187470, −4.85502898063728844739466830560, −3.67218944522848342314411371887, −3.08330236232313765921606709258, −1.27415075733749215024234761238,
1.27415075733749215024234761238, 3.08330236232313765921606709258, 3.67218944522848342314411371887, 4.85502898063728844739466830560, 5.85900984964464586455726187470, 6.57624001584826522381764449343, 7.71078313759863912854007134813, 8.312293789871667127278945468914, 9.566831202589947576418410084981, 10.30328332414120198497263720819
