| L(s) = 1 | + (1 − i)3-s + (1 + 2i)5-s + (−1 + i)7-s + i·9-s + 4·11-s + (3 + 3i)13-s + (3 + i)15-s + (−3 − 3i)17-s − 6i·19-s + 2i·21-s + (−3 − 3i)23-s + (−3 + 4i)25-s + (4 + 4i)27-s − 2·29-s − 6i·31-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.577i)3-s + (0.447 + 0.894i)5-s + (−0.377 + 0.377i)7-s + 0.333i·9-s + 1.20·11-s + (0.832 + 0.832i)13-s + (0.774 + 0.258i)15-s + (−0.727 − 0.727i)17-s − 1.37i·19-s + 0.436i·21-s + (−0.625 − 0.625i)23-s + (−0.600 + 0.800i)25-s + (0.769 + 0.769i)27-s − 0.371·29-s − 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67650 + 0.195202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67650 + 0.195202i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (3 + 3i)T + 23iT^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (9 - 9i)T - 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + (9 + 9i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (3 - 3i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (7 + 7i)T + 97iT^{2} \) |
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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
−11.42970449629801196089870221042, −11.02032927822570671658654264873, −9.462901133023270393262549780446, −9.071570556925653699035058463326, −7.76297179005460585182312075616, −6.70417239734258928387200378300, −6.21382746456286194991743353501, −4.43660479879381727218012065668, −2.96927320154170829308493493698, −1.93694329587974974996183938080,
1.44439507623810985111621831824, 3.52281069771939813450087708539, 4.16317567156087977885479179183, 5.71506287885804007776026299680, 6.56410494578771539097241739500, 8.203404459831289937590137128669, 8.801708660031791190087577861401, 9.717172452821239387655322594668, 10.33616522897688550485706945452, 11.68837005239998595916829500718
