| L(s) = 1 | − 1.73i·5-s + (−2 − 1.73i)7-s − 3.46i·11-s − 3.46i·13-s + 5.19i·17-s + 2·19-s − 6.92i·23-s + 2.00·25-s + 6·29-s + 10·31-s + (−2.99 + 3.46i)35-s − 11·37-s + 1.73i·41-s − 5.19i·43-s − 9·47-s + ⋯ |
| L(s) = 1 | − 0.774i·5-s + (−0.755 − 0.654i)7-s − 1.04i·11-s − 0.960i·13-s + 1.26i·17-s + 0.458·19-s − 1.44i·23-s + 0.400·25-s + 1.11·29-s + 1.79·31-s + (−0.507 + 0.585i)35-s − 1.80·37-s + 0.270i·41-s − 0.792i·43-s − 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.116076265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116076265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 1.73iT - 41T^{2} \) |
| 43 | \( 1 + 5.19iT - 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 15T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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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
−8.323066746112530379837904426301, −7.948871936717258972109326072486, −6.56413915701137997547852606466, −6.34139643759429109096478155328, −5.23722392792882955840296559326, −4.53405262126083068144133957227, −3.50008211691263269420735732445, −2.88416416133051300556681451058, −1.27100993785628968955229239785, −0.37133089278386498690006053141,
1.56791815460575822623627054338, 2.74237400063408006018306758615, 3.21055294510375816989915517161, 4.47954177623568957934762654334, 5.16103961017549742741081044620, 6.21591285797117636573162204166, 6.90651622641196042568144878753, 7.24724739595439216965252383462, 8.366494203928348143144652794566, 9.200042275059527863904609124796
