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
−10.66201199161312921998886303279, −9.880480280114892180893248021901, −9.256838209890321935467360807473, −8.816220291104302378137142787199, −7.52166898856162210942310686422, −5.84763064941362546848696784863, −5.12188648414183212959296784722, −4.34830857532347625125336907580, −2.78343948186860015235716336647, −2.02166679814079099484046829008,
0.874620487700127789595186109791, 2.59831180681730077909049536057, 3.80826456301963973685822028475, 5.43758100834088969365253709505, 6.17873245145297124364570736141, 7.16836714918167322698103097687, 7.54793776701561679018477057122, 8.732643142963682322308619115480, 9.528096378898595961454926296180, 10.62839843747414029970862921554