| L(s) = 1 | + i·3-s + (1 − 2i)5-s − 9-s + 2i·13-s + (2 + i)15-s − 2i·17-s − 8i·23-s + (−3 − 4i)25-s − i·27-s − 6·29-s − 2·31-s − 2·39-s − 6·41-s + 4i·43-s + (−1 + 2i)45-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.447 − 0.894i)5-s − 0.333·9-s + 0.554i·13-s + (0.516 + 0.258i)15-s − 0.485i·17-s − 1.66i·23-s + (−0.600 − 0.800i)25-s − 0.192i·27-s − 1.11·29-s − 0.359·31-s − 0.320·39-s − 0.937·41-s + 0.609i·43-s + (−0.149 + 0.298i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.053477162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.053477162\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1 + 2i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.383548701314581094005738613236, −7.58799807174121754531457809412, −6.59864227832452420350382859659, −5.92200057454183166228681597540, −5.04809116396817580826028248407, −4.55514749948917381427634365193, −3.72592233661079609108828597959, −2.60058697958479167476900121664, −1.65118825847638657844829718062, −0.28632232622969138591196146024,
1.40008303662157334188651310330, 2.23147212476362865354741925610, 3.21480349242437974936597189016, 3.85021290271186385778929707335, 5.25787449044516256193126391813, 5.78228332956048009940536469712, 6.47518856034023855536807133386, 7.41005828791500526273809370256, 7.59987583417597476609641885464, 8.671880058461965141559762387718
