| L(s) = 1 | − 4·11-s + 2·13-s + 2·17-s − 4·19-s + 2·29-s + 10·37-s − 10·41-s + 4·43-s − 8·47-s − 7·49-s − 10·53-s − 4·59-s − 2·61-s + 12·67-s − 8·71-s − 10·73-s − 12·83-s + 6·89-s − 2·97-s − 6·101-s − 16·103-s + 12·107-s + 14·109-s + 2·113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.371·29-s + 1.64·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 49-s − 1.37·53-s − 0.520·59-s − 0.256·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s − 1.31·83-s + 0.635·89-s − 0.203·97-s − 0.597·101-s − 1.57·103-s + 1.16·107-s + 1.34·109-s + 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.132479360823444390461948599740, −7.62818581639541327480210566204, −6.59119797171035359138537025559, −6.00095562582224088382030415028, −5.10854764303702027975036605471, −4.42469036791545275362320561796, −3.36550561270118716838366308509, −2.59055487455418232206827677965, −1.46780416341541155024498465553, 0,
1.46780416341541155024498465553, 2.59055487455418232206827677965, 3.36550561270118716838366308509, 4.42469036791545275362320561796, 5.10854764303702027975036605471, 6.00095562582224088382030415028, 6.59119797171035359138537025559, 7.62818581639541327480210566204, 8.132479360823444390461948599740
