| L(s) = 1 | + (1.40 + 1.01i)3-s + 5-s + 3.61i·7-s + (0.944 + 2.84i)9-s + 1.04·11-s + 0.0250i·13-s + (1.40 + 1.01i)15-s − 1.83i·17-s + 3.45·19-s + (−3.66 + 5.07i)21-s − 6.66i·23-s + 25-s + (−1.56 + 4.95i)27-s + 5.90i·29-s + 1.86i·31-s + ⋯ |
| L(s) = 1 | + (0.810 + 0.585i)3-s + 0.447·5-s + 1.36i·7-s + (0.314 + 0.949i)9-s + 0.314·11-s + 0.00694i·13-s + (0.362 + 0.261i)15-s − 0.443i·17-s + 0.793·19-s + (−0.799 + 1.10i)21-s − 1.38i·23-s + 0.200·25-s + (−0.300 + 0.953i)27-s + 1.09i·29-s + 0.334i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.942551639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.942551639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.40 - 1.01i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-5.81 - 5.76i)T \) |
| good | 7 | \( 1 - 3.61iT - 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 0.0250iT - 13T^{2} \) |
| 17 | \( 1 + 1.83iT - 17T^{2} \) |
| 19 | \( 1 - 3.45T + 19T^{2} \) |
| 23 | \( 1 + 6.66iT - 23T^{2} \) |
| 29 | \( 1 - 5.90iT - 29T^{2} \) |
| 31 | \( 1 - 1.86iT - 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.93T + 41T^{2} \) |
| 43 | \( 1 - 11.4iT - 43T^{2} \) |
| 47 | \( 1 + 4.63iT - 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 - 3.59iT - 59T^{2} \) |
| 61 | \( 1 - 6.07iT - 61T^{2} \) |
| 71 | \( 1 - 1.63iT - 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 5.00iT - 79T^{2} \) |
| 83 | \( 1 - 6.25iT - 83T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
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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
−8.799132718949794207679598626248, −8.170907222324456509874919740563, −7.26170621044210462061099008794, −6.38704943104180085030026318700, −5.54394643862559116234164836079, −4.93075009438573136488847060765, −4.08706680857255879630706998351, −2.84957232336635323284962784965, −2.62948701836019711519852300170, −1.41872920438129121105170076096,
0.78698241813485584171035314054, 1.61097365202263977416852791736, 2.64267772429186719684750628372, 3.70996491361333436475383648096, 4.08425510878134476757560608546, 5.32845209156411895060198586783, 6.23254742397054144923788991668, 6.90116085264096031714773326524, 7.63732436083781294120540186849, 7.971939631966359008693663532156
