L(s) = 1 | + 1.41i·2-s + (−1.58 + 0.707i)3-s − 2.00·4-s + 2.23i·5-s + (−1.00 − 2.23i)6-s + 3.16·7-s − 2.82i·8-s + (2.00 − 2.23i)9-s − 3.16·10-s + (3.16 − 1.41i)12-s + 4.47i·14-s + (−1.58 − 3.53i)15-s + 4.00·16-s + (3.16 + 2.82i)18-s − 4.47i·20-s + (−5.00 + 2.23i)21-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + (−0.912 + 0.408i)3-s − 1.00·4-s + 0.999i·5-s + (−0.408 − 0.912i)6-s + 1.19·7-s − 1.00i·8-s + (0.666 − 0.745i)9-s − 1.00·10-s + (0.912 − 0.408i)12-s + 1.19i·14-s + (−0.408 − 0.912i)15-s + 1.00·16-s + (0.745 + 0.666i)18-s − 1.00i·20-s + (−1.09 + 0.487i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.374372 + 0.577529i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.374372 + 0.577529i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (1.58 - 0.707i)T \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−15.31526433221270455487829857567, −14.75183761004493719035284875953, −13.58696131546888214264784967849, −11.93361323158611850294817609907, −10.86350805723286984809499761141, −9.747336975782428837565256190701, −8.057232819118948939437920310797, −6.81832180906283164271842194157, −5.61045984548333734292758686337, −4.24345971898683301440855894128,
1.48983611877484028104485383038, 4.52760392352707541403596359059, 5.47125061701447773016299072657, 7.76814223664752297023615196726, 8.988021565473084263259272139074, 10.52909702441978037345575630526, 11.49501306157240950308715152306, 12.32212370350680719602150609768, 13.23077157856824661009931669758, 14.40884061334837374941686196439