| L(s) = 1 | − 3·3-s − 12·7-s + 9·9-s − 20·11-s + 58·13-s + 70·17-s − 92·19-s + 36·21-s − 112·23-s − 27·27-s + 66·29-s − 108·31-s + 60·33-s + 58·37-s − 174·39-s + 66·41-s + 388·43-s + 408·47-s − 199·49-s − 210·51-s − 474·53-s + 276·57-s − 540·59-s + 14·61-s − 108·63-s + 276·67-s + 336·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.647·7-s + 1/3·9-s − 0.548·11-s + 1.23·13-s + 0.998·17-s − 1.11·19-s + 0.374·21-s − 1.01·23-s − 0.192·27-s + 0.422·29-s − 0.625·31-s + 0.316·33-s + 0.257·37-s − 0.714·39-s + 0.251·41-s + 1.37·43-s + 1.26·47-s − 0.580·49-s − 0.576·51-s − 1.22·53-s + 0.641·57-s − 1.19·59-s + 0.0293·61-s − 0.215·63-s + 0.503·67-s + 0.586·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 474 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 14 T + p^{3} T^{2} \) |
| 67 | \( 1 - 276 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 790 T + p^{3} T^{2} \) |
| 79 | \( 1 - 308 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1426 T + p^{3} T^{2} \) |
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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
−8.121985779443298893593156447530, −7.53438384046935713194458120841, −6.31862999860305620436195919815, −6.12712457424435070958923373272, −5.18126597890979425062731125142, −4.13501554239209220345306052308, −3.42554146853679706859891586572, −2.26606097685825236268782882592, −1.06663966272672750298423172366, 0,
1.06663966272672750298423172366, 2.26606097685825236268782882592, 3.42554146853679706859891586572, 4.13501554239209220345306052308, 5.18126597890979425062731125142, 6.12712457424435070958923373272, 6.31862999860305620436195919815, 7.53438384046935713194458120841, 8.121985779443298893593156447530
