| L(s) = 1 | − 1.41i·5-s − i·7-s + 2.82·11-s − 7.07i·17-s − 4i·19-s − 2.82·23-s + 2.99·25-s + 4.24i·29-s − 4i·31-s − 1.41·35-s − 2·37-s − 4.24i·41-s + 12i·43-s − 5.65·47-s − 49-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s − 0.377i·7-s + 0.852·11-s − 1.71i·17-s − 0.917i·19-s − 0.589·23-s + 0.599·25-s + 0.787i·29-s − 0.718i·31-s − 0.239·35-s − 0.328·37-s − 0.662i·41-s + 1.82i·43-s − 0.825·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527606252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527606252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.258130291003525277409524725592, −7.34532869817198431800479167919, −6.84762062586176595234387870908, −6.00087649347226538195786724932, −4.94066657179592960634606639914, −4.61277542768498249466778608768, −3.55666544793019625409964544010, −2.66666705025501343148272313238, −1.43261523446288966640833569152, −0.45162906793302675177995093192,
1.40549860883651071583469321700, 2.24957442573755132810561704388, 3.43624553982080252355332338352, 3.92037559336775291813691583620, 4.96082019723214938159232688179, 6.07590455493688159477801507928, 6.27292048356445717471348977429, 7.19510089833014848074146942697, 8.055551366895302176221553435119, 8.612093717791084408449634493180
