| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s − 10-s + 12-s − 15-s + 16-s − 20-s − 23-s + 24-s + 25-s − 27-s − 29-s − 30-s + 32-s − 40-s + 41-s − 43-s − 46-s − 2·47-s + 48-s + 50-s − 54-s − 58-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s − 10-s + 12-s − 15-s + 16-s − 20-s − 23-s + 24-s + 25-s − 27-s − 29-s − 30-s + 32-s − 40-s + 41-s − 43-s − 46-s − 2·47-s + 48-s + 50-s − 54-s − 58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.089952523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.089952523\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 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 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 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
−10.31990048532708246071458591738, −9.286706436347398469698404230018, −8.215826122258727690043498668495, −7.79212778268396863378710472128, −6.87674251740806942747160120192, −5.82575399864480327738129438243, −4.71051663907163530853475702508, −3.77607521606670155779079207040, −3.16147202131854107662615396429, −2.01168583171837733149191587769,
2.01168583171837733149191587769, 3.16147202131854107662615396429, 3.77607521606670155779079207040, 4.71051663907163530853475702508, 5.82575399864480327738129438243, 6.87674251740806942747160120192, 7.79212778268396863378710472128, 8.215826122258727690043498668495, 9.286706436347398469698404230018, 10.31990048532708246071458591738
