| L(s) = 1 | + 2-s + 4-s + (−1.88 − 1.20i)5-s + 8-s + (−1.88 − 1.20i)10-s − 1.59i·11-s − 0.925·13-s + 16-s − 3.95i·17-s + 0.625i·19-s + (−1.88 − 1.20i)20-s − 1.59i·22-s − 7.78·23-s + (2.08 + 4.54i)25-s − 0.925·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.841 − 0.539i)5-s + 0.353·8-s + (−0.595 − 0.381i)10-s − 0.480i·11-s − 0.256·13-s + 0.250·16-s − 0.959i·17-s + 0.143i·19-s + (−0.420 − 0.269i)20-s − 0.339i·22-s − 1.62·23-s + (0.416 + 0.908i)25-s − 0.181·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1801637227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1801637227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.88 + 1.20i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 1.59iT - 11T^{2} \) |
| 13 | \( 1 + 0.925T + 13T^{2} \) |
| 17 | \( 1 + 3.95iT - 17T^{2} \) |
| 19 | \( 1 - 0.625iT - 19T^{2} \) |
| 23 | \( 1 + 7.78T + 23T^{2} \) |
| 29 | \( 1 - 9.34iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 0.426iT - 37T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 - 6.27iT - 43T^{2} \) |
| 47 | \( 1 - 2.78iT - 47T^{2} \) |
| 53 | \( 1 - 3.35T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 0.356iT - 67T^{2} \) |
| 71 | \( 1 + 9.07iT - 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 + 0.809iT - 83T^{2} \) |
| 89 | \( 1 + 4.01T + 89T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−7.83218104420176067786032874807, −7.31258887262369189915666023053, −6.40826410210803822572044830752, −5.61503978511205971760289537263, −4.90347355330414300546159965170, −4.19065744708384992862787402002, −3.48161935451487718953948569690, −2.61061194020418979871430950608, −1.38154836561410912500299047523, −0.03707767072342838893985084988,
1.73586442821490192892490729923, 2.62506990996532227730066181671, 3.64570660274189739657405163577, 4.10152832751056938289913782632, 4.94183600063223941373467266984, 5.84597952049911949835276403327, 6.62661978062546428757183512935, 7.13657700193162574302915982141, 8.069484005910987634218933606081, 8.366295141215864082740588154772
