L(s) = 1 | + (−1 − 2i)5-s − 4i·7-s + 4·11-s + 4i·17-s − 4i·23-s + (−3 + 4i)25-s − 6·29-s + 4·31-s + (−8 + 4i)35-s + 8i·37-s + 10·41-s + 4i·43-s − 4i·47-s − 9·49-s + 12i·53-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.894i)5-s − 1.51i·7-s + 1.20·11-s + 0.970i·17-s − 0.834i·23-s + (−0.600 + 0.800i)25-s − 1.11·29-s + 0.718·31-s + (−1.35 + 0.676i)35-s + 1.31i·37-s + 1.56·41-s + 0.609i·43-s − 0.583i·47-s − 1.28·49-s + 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.934842 - 0.577764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.934842 - 0.577764i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−12.54147526348877983719269710054, −11.52645430686467146683683686669, −10.55163908123743082372848797502, −9.458454479382057127776008723656, −8.382882220249915634657648452507, −7.38886104694848865058450009257, −6.23353108455923494477092678570, −4.51725979962021635841320920932, −3.81805801479417206425639861289, −1.15980437545413181859700964876,
2.41444319834448939407476003740, 3.78070346781804821885924275094, 5.47840540371397839884954858337, 6.54357012282433405503891675422, 7.64201655135141947135104127120, 8.949372517253268092969552758100, 9.658139086457238696235799681109, 11.21489140015531959037072666364, 11.69464799482469199651062459216, 12.60025063249490897346720021931