| L(s) = 1 | − 0.153i·3-s − 2.94i·5-s + (−0.319 − 2.62i)7-s + 2.97·9-s + (1.01 − 3.15i)11-s + 13-s − 0.452·15-s − 5.02·17-s − 4.07·19-s + (−0.403 + 0.0490i)21-s − 6.95·23-s − 3.69·25-s − 0.917i·27-s − 5.04i·29-s − 0.575i·31-s + ⋯ |
| L(s) = 1 | − 0.0886i·3-s − 1.31i·5-s + (−0.120 − 0.992i)7-s + 0.992·9-s + (0.306 − 0.951i)11-s + 0.277·13-s − 0.116·15-s − 1.21·17-s − 0.935·19-s + (−0.0879 + 0.0107i)21-s − 1.45·23-s − 0.739·25-s − 0.176i·27-s − 0.936i·29-s − 0.103i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.207332987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207332987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.319 + 2.62i)T \) |
| 11 | \( 1 + (-1.01 + 3.15i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.153iT - 3T^{2} \) |
| 5 | \( 1 + 2.94iT - 5T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 + 5.04iT - 29T^{2} \) |
| 31 | \( 1 + 0.575iT - 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 + 4.51T + 41T^{2} \) |
| 43 | \( 1 + 5.12iT - 43T^{2} \) |
| 47 | \( 1 - 6.71iT - 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 59 | \( 1 - 13.5iT - 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 0.423T + 73T^{2} \) |
| 79 | \( 1 + 8.71iT - 79T^{2} \) |
| 83 | \( 1 - 9.18T + 83T^{2} \) |
| 89 | \( 1 + 4.93iT - 89T^{2} \) |
| 97 | \( 1 - 0.793iT - 97T^{2} \) |
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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
−8.187567731692223117082205937743, −7.40533320419392377006130121824, −6.49618989063016612675064000602, −5.98391535709923628779162651101, −4.82344613275778833587164058116, −4.18499186771559787497392851968, −3.80951072107812761839460759703, −2.21019609553283022326610075455, −1.23179246310848194119677844018, −0.34707614056627834193611480134,
1.88887961060589993099407102582, 2.30765826649833172109484713695, 3.46177812792360386119766526438, 4.21810403921517923348879677024, 5.03333272546867660624577350533, 6.18153245249865554823296391584, 6.66234116438992434525330159988, 7.12013602747187286014998763461, 8.138156378741970301127431660205, 8.797882890032191236169244408051
