| L(s) = 1 | + 2.53i·2-s − 2.40·4-s + 8.44·5-s + 2.44i·7-s + 4.02i·8-s + 21.3i·10-s + 11.7i·13-s − 6.17·14-s − 19.8·16-s − 2.02i·17-s + 8.47i·19-s − 20.3·20-s − 41.9·23-s + 46.3·25-s − 29.8·26-s + ⋯ |
| L(s) = 1 | + 1.26i·2-s − 0.602·4-s + 1.68·5-s + 0.348i·7-s + 0.503i·8-s + 2.13i·10-s + 0.905i·13-s − 0.441·14-s − 1.23·16-s − 0.119i·17-s + 0.445i·19-s − 1.01·20-s − 1.82·23-s + 1.85·25-s − 1.14·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.503592457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.503592457\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.53iT - 4T^{2} \) |
| 5 | \( 1 - 8.44T + 25T^{2} \) |
| 7 | \( 1 - 2.44iT - 49T^{2} \) |
| 13 | \( 1 - 11.7iT - 169T^{2} \) |
| 17 | \( 1 + 2.02iT - 289T^{2} \) |
| 19 | \( 1 - 8.47iT - 361T^{2} \) |
| 23 | \( 1 + 41.9T + 529T^{2} \) |
| 29 | \( 1 - 24.6iT - 841T^{2} \) |
| 31 | \( 1 - 21.8T + 961T^{2} \) |
| 37 | \( 1 + 16.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 49.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 118. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.41T + 4.48e3T^{2} \) |
| 71 | \( 1 + 6.03T + 5.04e3T^{2} \) |
| 73 | \( 1 + 6.08iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 30.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 60.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 36.6T + 9.40e3T^{2} \) |
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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
−9.865623846770544116838130116737, −9.033366412075007640358182176627, −8.469851655720648234403866113912, −7.33630915750178127643465157466, −6.56476675952243103481752188616, −5.87371406340235806511061035380, −5.41930416314319804667111778672, −4.27613432863398362391400998620, −2.51969499817948238393305403693, −1.72035599350717429639966849250,
0.71413323541313740925684957657, 1.89540056603917743225799742406, 2.55731196507463697397923456743, 3.66142384984693094200494535066, 4.81684969405205471931512357132, 5.90335922351122111951120600590, 6.51050811342059227805709122775, 7.73084304410416515106257035912, 8.859003271629383418086782290710, 9.715412674576740604893869931967
