L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.258 + 0.965i)11-s + (−1 − i)13-s + (−0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.36 + 0.366i)19-s − i·22-s + (−0.707 + 1.22i)23-s + (−0.866 + 0.5i)25-s + (1.22 + 0.707i)26-s + 28-s + (−0.707 + 0.707i)29-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.258 + 0.965i)11-s + (−1 − i)13-s + (−0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (−1.36 + 0.366i)19-s − i·22-s + (−0.707 + 1.22i)23-s + (−0.866 + 0.5i)25-s + (1.22 + 0.707i)26-s + 28-s + (−0.707 + 0.707i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5040466158\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5040466158\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 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 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.517 + 1.93i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.150711174870492199809719862312, −8.196602034472543332718558180172, −7.925502430356426879537732398426, −7.14793613232703863376227694125, −6.30495011740410918737160568909, −5.29673500669524120284276370376, −4.92966034015020175278673285929, −3.46167438337546513749952591241, −2.23320531826477562168352594061, −1.67682509727456394218402982047,
0.39140652143569007224823501915, 1.97558138506320109132920793586, 2.48900880462101333229988881673, 3.96703588176040379938055162294, 4.47739056072834068925645726941, 5.82920677077697369907656607382, 6.50976316471980796764610321760, 7.38236756097404828319770879112, 8.009495712852487821556980355469, 8.581613039467562329099639121642