| L(s) = 1 | + 5·3-s + 2·7-s − 2·9-s − 39·11-s − 84·13-s + 61·17-s − 151·19-s + 10·21-s − 58·23-s − 145·27-s + 192·29-s + 18·31-s − 195·33-s + 138·37-s − 420·39-s + 229·41-s − 164·43-s − 212·47-s − 339·49-s + 305·51-s − 578·53-s − 755·57-s + 336·59-s + 858·61-s − 4·63-s − 209·67-s − 290·69-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.107·7-s − 0.0740·9-s − 1.06·11-s − 1.79·13-s + 0.870·17-s − 1.82·19-s + 0.103·21-s − 0.525·23-s − 1.03·27-s + 1.22·29-s + 0.104·31-s − 1.02·33-s + 0.613·37-s − 1.72·39-s + 0.872·41-s − 0.581·43-s − 0.657·47-s − 0.988·49-s + 0.837·51-s − 1.49·53-s − 1.75·57-s + 0.741·59-s + 1.80·61-s − 0.00799·63-s − 0.381·67-s − 0.505·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 13 | \( 1 + 84 T + p^{3} T^{2} \) |
| 17 | \( 1 - 61 T + p^{3} T^{2} \) |
| 19 | \( 1 + 151 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 192 T + p^{3} T^{2} \) |
| 31 | \( 1 - 18 T + p^{3} T^{2} \) |
| 37 | \( 1 - 138 T + p^{3} T^{2} \) |
| 41 | \( 1 - 229 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 212 T + p^{3} T^{2} \) |
| 53 | \( 1 + 578 T + p^{3} T^{2} \) |
| 59 | \( 1 - 336 T + p^{3} T^{2} \) |
| 61 | \( 1 - 858 T + p^{3} T^{2} \) |
| 67 | \( 1 + 209 T + p^{3} T^{2} \) |
| 71 | \( 1 - 780 T + p^{3} T^{2} \) |
| 73 | \( 1 - 403 T + p^{3} T^{2} \) |
| 79 | \( 1 - 230 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1293 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1369 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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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
−10.14414494571919100377156405929, −9.592344845456475752492739496436, −8.224689310321065489130005056983, −8.007946209039481790518386262374, −6.75885525509895332443465875943, −5.42191351455452004053060582519, −4.38475705027047916005275487976, −2.92202635500835945280795686299, −2.20007281798438540565244237341, 0,
2.20007281798438540565244237341, 2.92202635500835945280795686299, 4.38475705027047916005275487976, 5.42191351455452004053060582519, 6.75885525509895332443465875943, 8.007946209039481790518386262374, 8.224689310321065489130005056983, 9.592344845456475752492739496436, 10.14414494571919100377156405929
