L(s) = 1 | + (−0.5 − 0.866i)3-s + (1 − 1.73i)4-s − 5-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s − 1.99·12-s + (2.5 + 2.59i)13-s + (0.5 + 0.866i)15-s + (−1.99 − 3.46i)16-s + (2 − 3.46i)19-s + (−1 + 1.73i)20-s − 0.999·21-s + (3 + 5.19i)23-s + 25-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.5 − 0.866i)4-s − 0.447·5-s + (0.188 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s − 0.577·12-s + (0.693 + 0.720i)13-s + (0.129 + 0.223i)15-s + (−0.499 − 0.866i)16-s + (0.458 − 0.794i)19-s + (−0.223 + 0.387i)20-s − 0.218·21-s + (0.625 + 1.08i)23-s + 0.200·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.759673 - 0.769478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.759673 - 0.769478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (-2.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 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
−12.02541047380748105871975083181, −11.02829777530720969891590156505, −10.84986654853285020380290885023, −9.261166812862292847696296958717, −8.091813077184630910476529489535, −7.00038096993446722771883789148, −6.01542497157830363601668527906, −4.96211734526200313205494333773, −3.04449626058766957781459448300, −1.07379306920934022469609976083,
2.58109931944481311541709170597, 3.98433022131580968741904523504, 5.17626494180446787848955615202, 6.66515327770735790034380105808, 7.78403186846411373020447714006, 8.530734226275003310524191406549, 10.03798448190038015015863148124, 10.78683824039766151770167721586, 11.98933576687973113600939460121, 12.40051934206655345615266105142