| L(s) = 1 | − i·2-s + i·3-s − 4-s + (1 + 2i)5-s + 6-s − i·7-s + i·8-s − 9-s + (2 − i)10-s + 5·11-s − i·12-s − 14-s + (−2 + i)15-s + 16-s − i·17-s + i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.447 + 0.894i)5-s + 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (0.632 − 0.316i)10-s + 1.50·11-s − 0.288i·12-s − 0.267·14-s + (−0.516 + 0.258i)15-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.734509122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734509122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - iT - 97T^{2} \) |
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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
−9.898806548509306387902157028838, −9.432365744164382493535185519442, −8.554724625608482015802809910950, −7.34736670098169758450464632897, −6.50567474779619146881879836596, −5.60384256760759524094402033659, −4.38541092871708392793850025358, −3.63997426744686948500345781353, −2.71631083381843678212150511518, −1.35581878153588960545147587834,
0.895888343365722567060548398489, 2.12256468552869514027262606471, 3.78019112165209341356467054099, 4.75956057505067696990028294807, 5.72328564930003353726158881728, 6.40974912693474023525917083813, 7.13639854461668622546552540165, 8.346948234449979839643579756319, 8.742533599501486685955799282245, 9.461932428575362119272997614415
