L(s) = 1 | + (1.35 − 0.394i)2-s + (−0.707 + 0.707i)3-s + (1.68 − 1.07i)4-s + (−1.75 + 1.38i)5-s + (−0.681 + 1.23i)6-s + (−2.47 − 2.47i)7-s + (1.87 − 2.11i)8-s − 1.00i·9-s + (−1.83 + 2.57i)10-s + 3.02i·11-s + (−0.437 + 1.95i)12-s + (0.363 + 0.363i)13-s + (−4.34 − 2.38i)14-s + (0.256 − 2.22i)15-s + (1.70 − 3.61i)16-s + (−2.36 + 2.36i)17-s + ⋯ |
L(s) = 1 | + (0.960 − 0.278i)2-s + (−0.408 + 0.408i)3-s + (0.844 − 0.535i)4-s + (−0.783 + 0.621i)5-s + (−0.278 + 0.505i)6-s + (−0.936 − 0.936i)7-s + (0.661 − 0.749i)8-s − 0.333i·9-s + (−0.579 + 0.814i)10-s + 0.913i·11-s + (−0.126 + 0.563i)12-s + (0.100 + 0.100i)13-s + (−1.16 − 0.638i)14-s + (0.0663 − 0.573i)15-s + (0.426 − 0.904i)16-s + (−0.573 + 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13211 - 0.0588211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13211 - 0.0588211i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 + 0.394i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.75 - 1.38i)T \) |
good | 7 | \( 1 + (2.47 + 2.47i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.02iT - 11T^{2} \) |
| 13 | \( 1 + (-0.363 - 0.363i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.36 - 2.36i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 + (-0.900 + 0.900i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.50iT - 29T^{2} \) |
| 31 | \( 1 - 3.85iT - 31T^{2} \) |
| 37 | \( 1 + (0.363 - 0.363i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.72T + 41T^{2} \) |
| 43 | \( 1 + (3.92 - 3.92i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.85 + 5.85i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.14 - 3.14i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + (-3.92 - 3.92i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.25iT - 71T^{2} \) |
| 73 | \( 1 + (-9.28 - 9.28i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.399T + 79T^{2} \) |
| 83 | \( 1 + (-0.199 + 0.199i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.28iT - 89T^{2} \) |
| 97 | \( 1 + (-6.73 + 6.73i)T - 97iT^{2} \) |
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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
−15.14842204528183520260688574443, −13.97020667312586228184528747230, −12.80726443676120092771426974029, −11.75249915797600334180292304126, −10.68976027580865864514972331879, −9.847185577527207281079643655937, −7.35642846402922270814853054122, −6.42737634161125575354669114117, −4.53011808042202940201052835960, −3.38477227989812797661300606691,
3.23476554326180866955618992189, 5.12045987168770241831574206311, 6.26972344492416972489918848919, 7.66632211036182986470108667259, 9.043323684358846287950424396288, 11.18726220073307894381233053579, 12.00324800912537837407477218774, 12.85022802355210355245258644241, 13.77488434072548104553198607366, 15.36790760819187953318912256821