| L(s) = 1 | − 1.60·3-s − 5-s + 4.33·7-s − 0.439·9-s − 0.439·13-s + 1.60·15-s − 7.10·19-s − 6.93·21-s − 3.90·23-s + 25-s + 5.50·27-s + 4·29-s + 8.22·31-s − 4.33·35-s + 0.878·37-s + 0.703·39-s + 10.8·41-s − 10.4·43-s + 0.439·45-s − 0.478·47-s + 11.7·49-s − 1.63·53-s + 11.3·57-s − 2.70·59-s − 12.0·61-s − 1.90·63-s + 0.439·65-s + ⋯ |
| L(s) = 1 | − 0.923·3-s − 0.447·5-s + 1.63·7-s − 0.146·9-s − 0.121·13-s + 0.413·15-s − 1.62·19-s − 1.51·21-s − 0.813·23-s + 0.200·25-s + 1.05·27-s + 0.742·29-s + 1.47·31-s − 0.732·35-s + 0.144·37-s + 0.112·39-s + 1.69·41-s − 1.58·43-s + 0.0654·45-s − 0.0698·47-s + 1.68·49-s − 0.225·53-s + 1.50·57-s − 0.351·59-s − 1.54·61-s − 0.239·63-s + 0.0544·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230033050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230033050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 7 | \( 1 - 4.33T + 7T^{2} \) |
| 13 | \( 1 + 0.439T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 - 0.878T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 0.478T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 + 2.70T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 5.67T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.274983366193077892764975994693, −7.71933588298364494649727010457, −6.63953727805311466282731382150, −6.14680920330243448100409146574, −5.22905281904583934018479950567, −4.62383796215616544773848520052, −4.13349849800955985285701864729, −2.74824396326418187024260657131, −1.79334623356679325971751988077, −0.64570940866442085548828179891,
0.64570940866442085548828179891, 1.79334623356679325971751988077, 2.74824396326418187024260657131, 4.13349849800955985285701864729, 4.62383796215616544773848520052, 5.22905281904583934018479950567, 6.14680920330243448100409146574, 6.63953727805311466282731382150, 7.71933588298364494649727010457, 8.274983366193077892764975994693
