| L(s) = 1 | + 3-s + (−1 + 2i)5-s − 2i·7-s + 9-s − 2i·11-s − 2·13-s + (−1 + 2i)15-s + 6i·17-s − 8i·19-s − 2i·21-s + 4i·23-s + (−3 − 4i)25-s + 27-s + 8i·29-s − 2i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.447 + 0.894i)5-s − 0.755i·7-s + 0.333·9-s − 0.603i·11-s − 0.554·13-s + (−0.258 + 0.516i)15-s + 1.45i·17-s − 1.83i·19-s − 0.436i·21-s + 0.834i·23-s + (−0.600 − 0.800i)25-s + 0.192·27-s + 1.48i·29-s − 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.046157128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046157128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 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
−8.472265957283386840669805876342, −7.66990402823539862264235295878, −7.17608752393453152373898427159, −6.54939893834362399380307207137, −5.60257831360239646988579279010, −4.48099107336196954487092116749, −3.81550548915274542901625141548, −3.07125601519601540624737078443, −2.26233135901830134180366625644, −0.846455618909862638156068365217,
0.76059464814911069789425674428, 2.10652803904730204309678777279, 2.74427696285614815105492608089, 4.02800632675067231706846133713, 4.49273717386838013180738857005, 5.43489347138969183726316710585, 6.09222664372060925189475261887, 7.32885717521294159283979550907, 7.73703881860251520331600841664, 8.454161731621803197736000328595
