| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8512332682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8512332682\) |
| \(L(\frac{3}{2})\) |
|
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
−7.80563257372407776790170615246, −7.32234695737692488943411515672, −6.79728852527573995860923837539, −5.97159816439665565761875093438, −5.00677071619976542218364378348, −4.45340263796384186470463035718, −3.53485400377882928683687687805, −2.40676882871947681763154551739, −1.09171942325976820388126882099, −0.34429096829302063909031841128,
1.21989450950092372855963138197, 2.60346428382339155917303372024, 3.42629133145031281224156327350, 4.40399459001687952364358735953, 5.08841104091111109149699307885, 5.92843288253024878791015441465, 6.17991284314720756210997386592, 7.44194923429415247027638687525, 8.001948888705138735692138847468, 8.950666366484910787336237029627
