| L(s) = 1 | + i·2-s − 4-s − 2i·7-s − i·8-s − 3·11-s + 5i·13-s + 2·14-s + 16-s − 6i·17-s + 4·19-s − 3i·22-s − 3i·23-s − 5·26-s + 2i·28-s + 2·31-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s − 0.904·11-s + 1.38i·13-s + 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.917·19-s − 0.639i·22-s − 0.625i·23-s − 0.980·26-s + 0.377i·28-s + 0.359·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.244232520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244232520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 11iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−9.420525679454772877203034511728, −8.768144426448402497612075497874, −7.58794727800193626063451260072, −7.27513999484760146643073520654, −6.39921490773248249275584129978, −5.29955803600445732301081714029, −4.61873774142482044172142400452, −3.64157779589735690008606650510, −2.32132539554190224221488278781, −0.56734120648111237565321687509,
1.25343878722026111293291772392, 2.63712958415216055652487296407, 3.26572737579231381317378936959, 4.53361205237041627778382251288, 5.53345640713980372357263929244, 6.00472084189028754069493715054, 7.54082653014497995570031676082, 8.143027641339401129326045529882, 8.875818126978925739268733224242, 9.912404313503682969465533051733
