| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 14-s + 16-s − 20-s − 23-s + 25-s + 28-s + 29-s + 32-s − 35-s − 40-s + 41-s − 2·43-s − 46-s − 47-s + 50-s + 56-s + 58-s − 61-s + 64-s + 67-s − 70-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 14-s + 16-s − 20-s − 23-s + 25-s + 28-s + 29-s + 32-s − 35-s − 40-s + 41-s − 2·43-s − 46-s − 47-s + 50-s + 56-s + 58-s − 61-s + 64-s + 67-s − 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.999447962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.999447962\) |
| \(L(1)\) |
|
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 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.749654944033659244563806488017, −8.297955953500930311950522226139, −8.064602251537060799667998880619, −7.11572101307188272154874320635, −6.33670691617110057514805384076, −5.22967706730196066688472978901, −4.57558045053924770114119407947, −3.83373125599876798986069514037, −2.82042326800615590749525548111, −1.55283975946407401177895532739,
1.55283975946407401177895532739, 2.82042326800615590749525548111, 3.83373125599876798986069514037, 4.57558045053924770114119407947, 5.22967706730196066688472978901, 6.33670691617110057514805384076, 7.11572101307188272154874320635, 8.064602251537060799667998880619, 8.297955953500930311950522226139, 9.749654944033659244563806488017
