L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.5 + 0.866i)3-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.965 − 0.258i)6-s + (1.36 + 0.366i)7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.866 + 0.500i)10-s + (0.258 + 0.965i)11-s − i·12-s − i·13-s + 1.41i·14-s + (0.258 + 0.965i)15-s + (0.500 − 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.5 + 0.866i)3-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.965 − 0.258i)6-s + (1.36 + 0.366i)7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.866 + 0.500i)10-s + (0.258 + 0.965i)11-s − i·12-s − i·13-s + 1.41i·14-s + (0.258 + 0.965i)15-s + (0.500 − 0.866i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.451951227\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451951227\) |
\(L(1)\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T^{2} \) |
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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
−9.192447912122913617683603146004, −8.151050432056309050877312119373, −7.87540601754926808161247150075, −6.60794255342969947089736846039, −5.84988221845115677382913619003, −5.22323978210998485943813522267, −4.74032972564868080732922679726, −4.13975532319983302932636185282, −2.79098954376131619908999599629, −1.26445566648540390783227709204,
1.11332047790144720059780133828, 1.89512352713148096680430031183, 2.68848925415201388764206273139, 3.83320387706965072786074022327, 4.93763980718568091148210924710, 5.37527846858287344650146534124, 6.50743692509067266865668605188, 6.80942123362558161465928524880, 8.178367704021486533240498900324, 8.485063556415850417930026108071