| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s − 3·11-s + 3·13-s − 15-s + 3·17-s + 21-s + 23-s + 25-s − 5·27-s − 9·29-s + 2·31-s − 3·33-s − 35-s − 4·37-s + 3·39-s + 4·41-s − 2·43-s + 2·45-s + 13·47-s + 49-s + 3·51-s − 2·53-s + 3·55-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.832·13-s − 0.258·15-s + 0.727·17-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 0.359·31-s − 0.522·33-s − 0.169·35-s − 0.657·37-s + 0.480·39-s + 0.624·41-s − 0.304·43-s + 0.298·45-s + 1.89·47-s + 1/7·49-s + 0.420·51-s − 0.274·53-s + 0.404·55-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.58391649140171876112109893547, −7.40529132176188094712685538473, −6.04253214520663814704080313496, −5.64206459622421452000352744332, −4.76681620275260797538945231331, −3.84570637772852671923419021909, −3.20322727139404385910985105481, −2.44338712325920150195071041320, −1.37707876314464947136674312092, 0,
1.37707876314464947136674312092, 2.44338712325920150195071041320, 3.20322727139404385910985105481, 3.84570637772852671923419021909, 4.76681620275260797538945231331, 5.64206459622421452000352744332, 6.04253214520663814704080313496, 7.40529132176188094712685538473, 7.58391649140171876112109893547
