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
−9.461932428575362119272997614415, −8.742533599501486685955799282245, −8.346948234449979839643579756319, −7.13639854461668622546552540165, −6.40974912693474023525917083813, −5.72328564930003353726158881728, −4.75956057505067696990028294807, −3.78019112165209341356467054099, −2.12256468552869514027262606471, −0.895888343365722567060548398489,
1.35581878153588960545147587834, 2.71631083381843678212150511518, 3.63997426744686948500345781353, 4.38541092871708392793850025358, 5.60384256760759524094402033659, 6.50567474779619146881879836596, 7.34736670098169758450464632897, 8.554724625608482015802809910950, 9.432365744164382493535185519442, 9.898806548509306387902157028838