L(s) = 1 | + i·2-s + (0.5 − 1.65i)3-s − 4-s + (1.91 + 1.15i)5-s + (1.65 + 0.5i)6-s − 3.21i·7-s − i·8-s + (−2.5 − 1.65i)9-s + (−1.15 + 1.91i)10-s − 4.31i·11-s + (−0.5 + 1.65i)12-s − 3.31·13-s + 3.21·14-s + (2.87 − 2.59i)15-s + 16-s + 3.21·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.288 − 0.957i)3-s − 0.5·4-s + (0.855 + 0.518i)5-s + (0.677 + 0.204i)6-s − 1.21i·7-s − 0.353i·8-s + (−0.833 − 0.552i)9-s + (−0.366 + 0.604i)10-s − 1.30i·11-s + (−0.144 + 0.478i)12-s − 0.919·13-s + 0.860·14-s + (0.742 − 0.669i)15-s + 0.250·16-s + 0.780·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36892 - 0.727319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36892 - 0.727319i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.5 + 1.65i)T \) |
| 5 | \( 1 + (-1.91 - 1.15i)T \) |
| 19 | \( 1 + (4.31 + 0.605i)T \) |
good | 7 | \( 1 + 3.21iT - 7T^{2} \) |
| 11 | \( 1 + 4.31iT - 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 + 8.25T + 29T^{2} \) |
| 31 | \( 1 + 2.61iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.82iT - 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 1.81iT - 73T^{2} \) |
| 79 | \( 1 - 15.3iT - 79T^{2} \) |
| 83 | \( 1 + 6.43T + 83T^{2} \) |
| 89 | \( 1 + 2.61T + 89T^{2} \) |
| 97 | \( 1 - 3.36T + 97T^{2} \) |
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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
−10.62839843747414029970862921554, −9.528096378898595961454926296180, −8.732643142963682322308619115480, −7.54793776701561679018477057122, −7.16836714918167322698103097687, −6.17873245145297124364570736141, −5.43758100834088969365253709505, −3.80826456301963973685822028475, −2.59831180681730077909049536057, −0.874620487700127789595186109791,
2.02166679814079099484046829008, 2.78343948186860015235716336647, 4.34830857532347625125336907580, 5.12188648414183212959296784722, 5.84763064941362546848696784863, 7.52166898856162210942310686422, 8.816220291104302378137142787199, 9.256838209890321935467360807473, 9.880480280114892180893248021901, 10.66201199161312921998886303279