L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 2·7-s + 9-s + 2·10-s + 5·11-s + 2·12-s + 13-s + 4·14-s + 15-s − 4·16-s + 2·18-s + 2·19-s + 2·20-s + 2·21-s + 10·22-s − 6·23-s + 25-s + 2·26-s + 27-s + 4·28-s + 2·30-s − 31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1/3·9-s + 0.632·10-s + 1.50·11-s + 0.577·12-s + 0.277·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.436·21-s + 2.13·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.755·28-s + 0.365·30-s − 0.179·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.099901334\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.099901334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.965665581046466271253371617006, −7.22686880300659616368612415181, −6.34717619551649847485405554530, −5.92693601929046982192298193608, −5.10467549272398560063239939121, −4.15518206763275895766339619575, −3.99698828503142810142025023829, −2.94319755896905148160902021703, −2.11853120521585897391620849145, −1.22976389858324984109451917995,
1.22976389858324984109451917995, 2.11853120521585897391620849145, 2.94319755896905148160902021703, 3.99698828503142810142025023829, 4.15518206763275895766339619575, 5.10467549272398560063239939121, 5.92693601929046982192298193608, 6.34717619551649847485405554530, 7.22686880300659616368612415181, 7.965665581046466271253371617006