| L(s) = 1 | − i·2-s − 4-s − 3.59·5-s + 4.66·7-s + i·8-s + 3.59i·10-s + 0.902·11-s + 3.90i·13-s − 4.66i·14-s + 16-s + 3.88i·17-s + 2.00·19-s + 3.59·20-s − 0.902i·22-s − 9.11·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.60·5-s + 1.76·7-s + 0.353i·8-s + 1.13i·10-s + 0.272·11-s + 1.08i·13-s − 1.24i·14-s + 0.250·16-s + 0.943i·17-s + 0.460·19-s + 0.804·20-s − 0.192i·22-s − 1.90·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2917139487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2917139487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 223 | \( 1 + (-0.326 - 14.9i)T \) |
| good | 5 | \( 1 + 3.59T + 5T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 - 0.902T + 11T^{2} \) |
| 13 | \( 1 - 3.90iT - 13T^{2} \) |
| 17 | \( 1 - 3.88iT - 17T^{2} \) |
| 19 | \( 1 - 2.00T + 19T^{2} \) |
| 23 | \( 1 + 9.11T + 23T^{2} \) |
| 29 | \( 1 - 5.06iT - 29T^{2} \) |
| 31 | \( 1 + 7.64T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 + 1.40iT - 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 5.68iT - 47T^{2} \) |
| 53 | \( 1 + 12.3iT - 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 - 1.98iT - 61T^{2} \) |
| 67 | \( 1 + 13.9iT - 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 5.68iT - 79T^{2} \) |
| 83 | \( 1 - 2.04iT - 83T^{2} \) |
| 89 | \( 1 - 1.95iT - 89T^{2} \) |
| 97 | \( 1 + 10.0iT - 97T^{2} \) |
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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
−8.550521673745727760260375042461, −8.059542258489567302491776790179, −7.56315280466238141418480578705, −6.66180686935160273376508461715, −5.41351781663858720407296761167, −4.70177817365073373917938822429, −3.94230304298762911817784506824, −3.65740931694877202857754553735, −2.06543833484046397243507457168, −1.44892693120233640819597542407,
0.091159578765410569804714400462, 1.30018343770316506897963518003, 2.76209795989125394989854127978, 3.96524594712377235342697263222, 4.32655308100613250574646900292, 5.20913021750989237646896664274, 5.81580807627041149941556635338, 7.06242473538274004280107450592, 7.74900037174009215096257693634, 7.904731985827388272830525929003
