| L(s) = 1 | + (−1 − i)2-s + 2i·4-s + 2·7-s + (2 − 2i)8-s + 4i·13-s + (−2 − 2i)14-s − 4·16-s − 2·17-s + 4i·19-s + 4·23-s + (4 − 4i)26-s + 4i·28-s + 6i·29-s + 2·31-s + (4 + 4i)32-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s + 0.755·7-s + (0.707 − 0.707i)8-s + 1.10i·13-s + (−0.534 − 0.534i)14-s − 16-s − 0.485·17-s + 0.917i·19-s + 0.834·23-s + (0.784 − 0.784i)26-s + 0.755i·28-s + 1.11i·29-s + 0.359·31-s + (0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061652616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061652616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.320790409193191666809960371292, −8.718974671665002238263615243170, −7.997116646795624189080178291321, −7.18762160895183733870766140314, −6.42455766701994764882307435573, −5.06825763634671756864950316409, −4.28819733187227740104508553205, −3.33701481222936830189558629981, −2.13721464765839697552313211034, −1.30826941777821373899527253154,
0.53683360102902637325595730137, 1.83315656798907699820694681813, 3.06439080447701807611507163097, 4.65970643095780117224783665133, 5.08658340089497564463925497300, 6.14566749624004574361482338824, 6.86096902587961284899623245638, 7.78744452164894816009686588603, 8.269225860070414740493238278268, 9.020681192648695899950313126147
