L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s − i·7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)10-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (1.36 − 0.366i)19-s + (0.707 − 0.707i)20-s + (−0.707 + 1.22i)23-s + (0.499 + 0.866i)28-s + (0.707 − 0.707i)29-s + (1 + 1.73i)31-s + (0.965 + 0.258i)32-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s − i·7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)10-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (1.36 − 0.366i)19-s + (0.707 − 0.707i)20-s + (−0.707 + 1.22i)23-s + (0.499 + 0.866i)28-s + (0.707 − 0.707i)29-s + (1 + 1.73i)31-s + (0.965 + 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.040498140\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.040498140\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.872850538907561663186952610307, −7.900610221064241928259606357352, −7.54388725221362970308210368763, −7.01923412084864195377194828933, −6.11098434373134124853224552203, −5.22079798620560824540499291972, −4.34720961134143153385030864863, −3.75171179099940555968076289632, −2.97500778556655159749187244935, −0.924461253475204145556098802084,
0.877223318839565187864379131796, 2.32610254600102098701600992812, 2.99690401581498661889506370458, 4.09006979555540489156282376935, 4.57260389077395619629323381734, 5.62981627461139319286131673472, 6.15435683142829401444434070004, 7.52886162009592333582453488001, 8.185138618436716479440776960250, 8.813204227317979064646667411413