L(s) = 1 | − i·3-s − 9-s + 2i·13-s − 6i·17-s + 4·19-s + 8i·23-s + i·27-s + 2·29-s + 4·31-s − 10i·37-s + 2·39-s + 2·41-s − 4i·43-s − 8i·47-s + 7·49-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.333·9-s + 0.554i·13-s − 1.45i·17-s + 0.917·19-s + 1.66i·23-s + 0.192i·27-s + 0.371·29-s + 0.718·31-s − 1.64i·37-s + 0.320·39-s + 0.312·41-s − 0.609i·43-s − 1.16i·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.709257386\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.709257386\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.018574409208527807497150581216, −7.80793465112902674253802281958, −7.36169946822693015713506980255, −6.66486281027773215449425076838, −5.63862383356426440938832477893, −5.04767092684243523257376891517, −3.88204900476709232033875987577, −2.93848855213103300556524240822, −1.92618655284913238382515736032, −0.70254990911682462328982229892,
1.05715617158470588151172119113, 2.52994653065934908141082127596, 3.37883835183183365598732081906, 4.35770794430006288534649028740, 5.02205483332170177821919809679, 6.06402512889731506323991590050, 6.56160952910186902253650817422, 7.84968632164936963321705687795, 8.262392697611206204609273464165, 9.118162724489079567632692499530