| L(s) = 1 | + (−0.497 + 1.32i)2-s + (−1.50 − 1.31i)4-s − 1.25i·5-s + 7-s + (2.49 − 1.33i)8-s + (1.65 + 0.624i)10-s − 1.55i·11-s − 1.07i·13-s + (−0.497 + 1.32i)14-s + (0.528 + 3.96i)16-s − 0.0158·17-s − 2.35i·19-s + (−1.65 + 1.88i)20-s + (2.06 + 0.775i)22-s − 5.95·23-s + ⋯ |
| L(s) = 1 | + (−0.351 + 0.936i)2-s + (−0.752 − 0.658i)4-s − 0.560i·5-s + 0.377·7-s + (0.881 − 0.472i)8-s + (0.524 + 0.197i)10-s − 0.469i·11-s − 0.297i·13-s + (−0.133 + 0.353i)14-s + (0.132 + 0.991i)16-s − 0.00383·17-s − 0.539i·19-s + (−0.369 + 0.421i)20-s + (0.439 + 0.165i)22-s − 1.24·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9262378755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9262378755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.497 - 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 1.25iT - 5T^{2} \) |
| 11 | \( 1 + 1.55iT - 11T^{2} \) |
| 13 | \( 1 + 1.07iT - 13T^{2} \) |
| 17 | \( 1 + 0.0158T + 17T^{2} \) |
| 19 | \( 1 + 2.35iT - 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 + 0.469iT - 29T^{2} \) |
| 31 | \( 1 - 1.69T + 31T^{2} \) |
| 37 | \( 1 + 4.59iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 1.97iT - 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 + 1.86iT - 53T^{2} \) |
| 59 | \( 1 + 8.54iT - 59T^{2} \) |
| 61 | \( 1 + 3.92iT - 61T^{2} \) |
| 67 | \( 1 + 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 8.88iT - 83T^{2} \) |
| 89 | \( 1 - 0.240T + 89T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
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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
−9.151361643928831778720644613143, −8.347526728974102883201278372973, −7.965326563931488582968330520096, −6.89881278875915729812455253086, −6.14615048623035564249343042303, −5.21611833943150813739512904355, −4.63635193974363990279618087419, −3.47709932208345114257042253059, −1.78639506443631589969422330609, −0.42994867035734544951495645548,
1.45131256870593141120898723001, 2.42932936149613515478674991721, 3.47034965893330389781848324820, 4.35920244947254011644787919560, 5.25690169770811135668024984162, 6.50076068970241366338445690151, 7.36861088077420881098758933547, 8.212307669713270102365119755433, 8.828980792225812841110494640152, 10.03396515344661784197045079156
