| L(s) = 1 | + (1.02 + 1.71i)2-s + (−1.88 + 3.52i)4-s − 6.98i·5-s + 2.64i·7-s + (−7.99 + 0.383i)8-s + (11.9 − 7.18i)10-s − 6.55·11-s − 8.12i·13-s + (−4.53 + 2.71i)14-s + (−8.86 − 13.3i)16-s − 13.5·17-s − 20.4·19-s + (24.6 + 13.1i)20-s + (−6.73 − 11.2i)22-s − 20.8i·23-s + ⋯ |
| L(s) = 1 | + (0.513 + 0.857i)2-s + (−0.472 + 0.881i)4-s − 1.39i·5-s + 0.377i·7-s + (−0.998 + 0.0478i)8-s + (1.19 − 0.718i)10-s − 0.596·11-s − 0.625i·13-s + (−0.324 + 0.194i)14-s + (−0.554 − 0.832i)16-s − 0.798·17-s − 1.07·19-s + (1.23 + 0.659i)20-s + (−0.306 − 0.511i)22-s − 0.907i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0478 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0478 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8581852995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8581852995\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.02 - 1.71i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 6.98iT - 25T^{2} \) |
| 11 | \( 1 + 6.55T + 121T^{2} \) |
| 13 | \( 1 + 8.12iT - 169T^{2} \) |
| 17 | \( 1 + 13.5T + 289T^{2} \) |
| 19 | \( 1 + 20.4T + 361T^{2} \) |
| 23 | \( 1 + 20.8iT - 529T^{2} \) |
| 29 | \( 1 + 54.4iT - 841T^{2} \) |
| 31 | \( 1 - 38.6iT - 961T^{2} \) |
| 37 | \( 1 + 46.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 25.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 78.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.60iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 48.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 63.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 16.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 15.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 59.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 134.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 155.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 104.T + 9.40e3T^{2} \) |
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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
−10.42250002151534154004934104545, −9.201338316986986037553303273166, −8.505504315908453996599511652311, −7.981577440615007430265799499896, −6.66553475759019838070144041284, −5.68305184403527963983579064590, −4.88793636656589279661654031654, −4.08752162286345791115764591814, −2.44709541479201678547460711829, −0.27039657712772875653597867520,
1.90118097761726938405853094393, 2.97200464198844314590858307708, 3.93621836697451741038184289225, 5.07672293451967578403807543221, 6.38442299244505636101259385630, 6.97256002241669092627405977086, 8.334019045834686553798610220588, 9.529836096121702710652471901037, 10.31313702928165914677889210511, 11.04448374188119780014922893828
