| L(s) = 1 | + (−1.77 + 1.35i)5-s − 4.11i·7-s − 1.61·11-s + 3.40i·13-s + 1.06i·17-s + 19-s + 5.34i·23-s + (1.33 − 4.81i)25-s + 7.98·29-s + 3.64·31-s + (5.56 + 7.31i)35-s − 7.59i·37-s − 5.94·41-s − 9.22i·43-s − 7.16i·47-s + ⋯ |
| L(s) = 1 | + (−0.795 + 0.605i)5-s − 1.55i·7-s − 0.486·11-s + 0.943i·13-s + 0.257i·17-s + 0.229·19-s + 1.11i·23-s + (0.266 − 0.963i)25-s + 1.48·29-s + 0.654·31-s + (0.941 + 1.23i)35-s − 1.24i·37-s − 0.928·41-s − 1.40i·43-s − 1.04i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5044139734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5044139734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.77 - 1.35i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.11iT - 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 3.40iT - 13T^{2} \) |
| 17 | \( 1 - 1.06iT - 17T^{2} \) |
| 23 | \( 1 - 5.34iT - 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 + 7.59iT - 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 + 9.22iT - 43T^{2} \) |
| 47 | \( 1 + 7.16iT - 47T^{2} \) |
| 53 | \( 1 - 2.57iT - 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 9.11iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 15.2iT - 73T^{2} \) |
| 79 | \( 1 - 6.34T + 79T^{2} \) |
| 83 | \( 1 + 6.54iT - 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 7.29iT - 97T^{2} \) |
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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
−8.114027371095451369067783579138, −7.45203924160519087898942383492, −7.00301463949982950230155194175, −6.30926440475168397908316889524, −5.10151628285560772062774949135, −4.22315132064243463472121553410, −3.74223263663290809844745095699, −2.81686060262305886963747080514, −1.46807952822582009737909210380, −0.16521087053562095850742936691,
1.23578693739334271217359748885, 2.74470196586894722042377060590, 3.06672487906058611957567858828, 4.60758905599001804790399482912, 4.92089049660854882253349539220, 5.86864375138608808631244881830, 6.52741087811733337885374343499, 7.70385726569543052663804828905, 8.261786543457444122598092366076, 8.664351639650643311548519012879
