| L(s) = 1 | + (−0.5 + 1.32i)2-s + (−1.50 − 1.32i)4-s + 2.64i·7-s + (2.50 − 1.32i)8-s + 4·11-s + (−3.50 − 1.32i)14-s + (0.500 + 3.96i)16-s + (−2 + 5.29i)22-s + 5.29i·23-s − 5·25-s + (3.50 − 3.96i)28-s + 10.5i·29-s + (−5.50 − 1.32i)32-s + 10.5i·37-s + 12·43-s + (−6.00 − 5.29i)44-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.935i)2-s + (−0.750 − 0.661i)4-s + 0.999i·7-s + (0.883 − 0.467i)8-s + 1.20·11-s + (−0.935 − 0.353i)14-s + (0.125 + 0.992i)16-s + (−0.426 + 1.12i)22-s + 1.10i·23-s − 25-s + (0.661 − 0.749i)28-s + 1.96i·29-s + (−0.972 − 0.233i)32-s + 1.73i·37-s + 1.82·43-s + (−0.904 − 0.797i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.556933 + 0.924800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.556933 + 0.924800i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.5 - 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.29iT - 23T^{2} \) |
| 29 | \( 1 - 10.5iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 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
−11.20961909287264512335255494150, −9.967431838943688016049105126496, −9.214816801595176800766717039881, −8.625903801365671790590457509253, −7.57992102978520621740334554130, −6.59623147254894945359251602149, −5.80431869535159251952096146703, −4.85560911127170109792540992087, −3.53264502831397871108854111783, −1.59107872405238015477568672891,
0.806097709578990692063738973092, 2.31471174308058104799078855800, 3.89518728377610978818893024479, 4.31315715256686274165587589135, 5.99081122971328745865502444770, 7.21622080974161626884662623880, 8.036103551643299844878937876337, 9.121566261915184788051845778858, 9.791990885502652367938957204653, 10.68269158535820971643676472078
