L(s) = 1 | − 2.50i·2-s − i·3-s − 4.26·4-s − 0.980i·5-s − 2.50·6-s + 5.66i·8-s − 9-s − 2.45·10-s + (−2.59 − 2.06i)11-s + 4.26i·12-s + 0.941·13-s − 0.980·15-s + 5.66·16-s − 5.17·17-s + 2.50i·18-s − 2.54·19-s + ⋯ |
L(s) = 1 | − 1.76i·2-s − 0.577i·3-s − 2.13·4-s − 0.438i·5-s − 1.02·6-s + 2.00i·8-s − 0.333·9-s − 0.776·10-s + (−0.782 − 0.622i)11-s + 1.23i·12-s + 0.261·13-s − 0.253·15-s + 1.41·16-s − 1.25·17-s + 0.589i·18-s − 0.584·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1673304291\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1673304291\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.59 + 2.06i)T \) |
good | 2 | \( 1 + 2.50iT - 2T^{2} \) |
| 5 | \( 1 + 0.980iT - 5T^{2} \) |
| 13 | \( 1 - 0.941T + 13T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 + 0.758T + 23T^{2} \) |
| 29 | \( 1 - 8.74iT - 29T^{2} \) |
| 31 | \( 1 + 6.36iT - 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 6.73iT - 43T^{2} \) |
| 47 | \( 1 - 9.73iT - 47T^{2} \) |
| 53 | \( 1 + 4.49T + 53T^{2} \) |
| 59 | \( 1 - 14.7iT - 59T^{2} \) |
| 61 | \( 1 + 5.62T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 7.65iT - 79T^{2} \) |
| 83 | \( 1 + 6.28T + 83T^{2} \) |
| 89 | \( 1 - 2.38iT - 89T^{2} \) |
| 97 | \( 1 + 7.30iT - 97T^{2} \) |
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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
−8.742889164065955336255737314242, −8.321245314755239062684120944951, −7.11093294086223088824052743866, −5.98670535869674899017627978907, −4.97938379518277712453931613050, −4.19068263136178647935964861616, −3.09096721898110205233603792786, −2.31677171048916261026532761192, −1.26495114077537858841896695360, −0.06769308287412271256195205472,
2.44362346772132380348929892174, 3.85584103613406480912529254031, 4.71160051554138308863882859906, 5.27444711023439189032583293285, 6.46179503024350414597217033953, 6.68776046657056672854597093863, 7.81381971695608223361538783390, 8.340277660386632602623262658135, 9.142878288992309434556415335849, 9.896020758552934179350186268888