L(s) = 1 | + 0.662i·3-s + (1.31 + 1.81i)5-s + (−1.19 + 2.35i)7-s + 2.56·9-s − 3.09i·11-s − 4.66·13-s + (−1.19 + 0.868i)15-s − 2.04·17-s − 5.60·19-s + (−1.56 − 0.794i)21-s − 1.87·23-s + (−1.56 + 4.74i)25-s + 3.68i·27-s + 3.56·29-s − 8.74·31-s + ⋯ |
L(s) = 1 | + 0.382i·3-s + (0.586 + 0.810i)5-s + (−0.453 + 0.891i)7-s + 0.853·9-s − 0.932i·11-s − 1.29·13-s + (−0.309 + 0.224i)15-s − 0.496·17-s − 1.28·19-s + (−0.340 − 0.173i)21-s − 0.390·23-s + (−0.312 + 0.949i)25-s + 0.708i·27-s + 0.661·29-s − 1.57·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6939853857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6939853857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.31 - 1.81i)T \) |
| 7 | \( 1 + (1.19 - 2.35i)T \) |
good | 3 | \( 1 - 0.662iT - 3T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 + 3.70iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 0.290iT - 47T^{2} \) |
| 53 | \( 1 - 9.49iT - 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 9.65iT - 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 + 1.35iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−9.456651450842552440906833232300, −8.908362900563050852991268165802, −7.88675083450126442768918137000, −6.94042904268071362255032425961, −6.35155428266556788802398839561, −5.57962471097542233676460576498, −4.68431898890761819206478062997, −3.63612692361164743536919240366, −2.69235334590077655122049291340, −1.92724229335019667617033104366,
0.21563303626818577646985240739, 1.64590956733914958400401139755, 2.34355671312138156550480718089, 3.98961689549855026338145387043, 4.52542017518390012443109692017, 5.34750930688669383010975343335, 6.56735917000027310556325956649, 6.99649601965189659190288020262, 7.72086549511006438529424443012, 8.680353946655441547329739576285