L(s) = 1 | + (0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)5-s − 1.99·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.999 − 1.73i)10-s + (−0.5 + 0.866i)11-s + (−0.999 − 1.73i)12-s − 4·13-s + 3.99·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−2 − 3.46i)19-s + 1.99·20-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s − 0.816·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−0.150 + 0.261i)11-s + (−0.288 − 0.499i)12-s − 1.10·13-s + 1.03·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (−0.458 − 0.794i)19-s + 0.447·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4611107178\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4611107178\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7 + 12.1i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.919402579247381229650396525367, −8.848230347662015276069176611433, −8.140560210250584206848825010845, −7.21502769034506298934328176616, −6.22260749729063194198420768014, −5.16844740100516949021650657759, −4.62101221991804696586040385955, −4.10491695588182596053675766811, −2.54773971534484745377811429033, −0.20032619055164613425568765078,
1.40038414670975303335191805557, 2.62900969894857450786351264314, 3.61307303972398834293074101973, 4.80258119780136087905492633840, 5.84619022743674287554604338135, 6.61994063891904614593524159116, 7.37428265443749302439275153552, 8.114974726979999944136498737233, 9.393050801438464190609356132860, 10.24803667058770397795027370933