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
−8.485063556415850417930026108071, −8.178367704021486533240498900324, −6.80942123362558161465928524880, −6.50743692509067266865668605188, −5.37527846858287344650146534124, −4.93763980718568091148210924710, −3.83320387706965072786074022327, −2.68848925415201388764206273139, −1.89512352713148096680430031183, −1.11332047790144720059780133828,
1.26445566648540390783227709204, 2.79098954376131619908999599629, 4.13975532319983302932636185282, 4.74032972564868080732922679726, 5.22323978210998485943813522267, 5.84988221845115677382913619003, 6.60794255342969947089736846039, 7.87540601754926808161247150075, 8.151050432056309050877312119373, 9.192447912122913617683603146004