L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s + 2·9-s − 6·11-s + i·12-s − 5i·13-s − 14-s + 16-s + 3i·17-s − 2i·18-s − 19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s + 0.666·9-s − 1.80·11-s + 0.288i·12-s − 1.38i·13-s − 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.471i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.174752 + 0.740261i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.174752 + 0.740261i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.876017528067997263728010117759, −8.660162121413821916064949680429, −7.77612921283094186400315734487, −7.36968635601476475027561036474, −5.94588866048212482367291290365, −5.15246677623423518985594486824, −4.01447318220950203180659533846, −2.89482614234160538226307188203, −1.83976765163521167928633654641, −0.33900283762510085792816425473,
2.04124522009189983557643337644, 3.48119750726470909916394202472, 4.60657500121310258618406628634, 5.20581549300400466359991490832, 6.18205371221264052512287134121, 7.36655446060614346946881406783, 7.72984663965283997999488472376, 9.139184014635326537383955715586, 9.363353566030147453465344299148, 10.42424864242956759141114593136