| L(s) = 1 | + i·3-s + (−2 − i)5-s + 4i·7-s − 9-s − 6·11-s − 4i·13-s + (1 − 2i)15-s + i·17-s + 4·19-s − 4·21-s + (3 + 4i)25-s − i·27-s − 8·31-s − 6i·33-s + (4 − 8i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s + 1.51i·7-s − 0.333·9-s − 1.80·11-s − 1.10i·13-s + (0.258 − 0.516i)15-s + 0.242i·17-s + 0.917·19-s − 0.872·21-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 1.43·31-s − 1.04i·33-s + (0.676 − 1.35i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7428461426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7428461426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 + i)T \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.488833408456977798164319739793, −7.86472709554145595082779138478, −7.10385045124249251485518819160, −5.75246770382834125329484034290, −5.24652418898834037902008916608, −4.98252278343401706839877768699, −3.56239159754619581502026423702, −3.05959349059321708465635830368, −2.09507618256717879985524917372, −0.31118452905429425724121262752,
0.71641015920013960347561500273, 2.03874170574212076204878314596, 3.11269689796595543533091579706, 3.80452687640327960960874067766, 4.68377776964618358131080015948, 5.44390378130265429126290665004, 6.63666714191099095413237584908, 7.19300603601074668081586068108, 7.61543356751347906695285154231, 8.125766967185313491067710418058
