L(s) = 1 | + (0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 − 0.866i)4-s + (−1.49 − 0.866i)6-s + (1 − 1.73i)7-s − 0.999·8-s − 2.99·9-s + (−1.49 + 0.866i)12-s + (−2 − 3.46i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s + 6·17-s + (−1.49 + 2.59i)18-s − 7·19-s + (−2.99 − 1.73i)21-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 − 0.433i)4-s + (−0.612 − 0.353i)6-s + (0.377 − 0.654i)7-s − 0.353·8-s − 0.999·9-s + (−0.433 + 0.249i)12-s + (−0.554 − 0.960i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.45·17-s + (−0.353 + 0.612i)18-s − 1.60·19-s + (−0.654 − 0.377i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.253086 - 1.43532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.253086 - 1.43532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.57021510800665011692791231956, −10.27846725553952859944396915196, −8.716621750390633761285193056045, −7.939291179382744755355383843969, −7.00482223282437383387882383757, −5.90281402754987293962748121180, −4.89793932327320862923762674513, −3.49986925564631839877185963284, −2.27912390423606772154902281763, −0.834927215286127367792479834614,
2.51849519034468010214177899111, 3.91434509398070000055591273648, 4.80672030919431728779799726607, 5.67769869635210712620004877875, 6.66298818596799400334007289537, 8.020429779086631399442985567981, 8.712950734487194954332160072320, 9.625778802538188999736617137024, 10.47174748901983443790504414975, 11.65472466997933109799150388013