| L(s) = 1 | − 3-s + (−1.5 − 1.65i)5-s − 2·9-s + 3.31i·11-s + (1.5 + 1.65i)15-s + 9·23-s + (−0.5 + 4.97i)25-s + 5·27-s + 9.94i·31-s − 3.31i·33-s − 9.94i·37-s + (3 + 3.31i)45-s + 12·47-s + 7·49-s + 13.2i·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.670 − 0.741i)5-s − 0.666·9-s + 1.00i·11-s + (0.387 + 0.428i)15-s + 1.87·23-s + (−0.100 + 0.994i)25-s + 0.962·27-s + 1.78i·31-s − 0.577i·33-s − 1.63i·37-s + (0.447 + 0.494i)45-s + 1.75·47-s + 49-s + 1.82i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.822813 + 0.316923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.822813 + 0.316923i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.5 + 1.65i)T \) |
| 11 | \( 1 - 3.31iT \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.94iT - 31T^{2} \) |
| 37 | \( 1 + 9.94iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 - 16.5iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 9.94iT - 97T^{2} \) |
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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
−10.49750899900514164184627894364, −9.045662088689659480756202590920, −8.862313002896001452782072213065, −7.54264756619224719872674404325, −6.97075687650188501283210541550, −5.66366374483449195480238652612, −4.99551092522424226964629754608, −4.10546097631714454915953794284, −2.77844053938080197046968206459, −1.06203982991045346899642712354,
0.57308266221351425213032390115, 2.69755607106233272800873131763, 3.51283509510263736619209835126, 4.75970576791150073815607932632, 5.79960484808337311986476830026, 6.51077679115038139650297938396, 7.44316089695929296856238710178, 8.338834303742206967607410489226, 9.104626914114269003628583249678, 10.32467223226139204382641656429
