| 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
−10.42424864242956759141114593136, −9.363353566030147453465344299148, −9.139184014635326537383955715586, −7.72984663965283997999488472376, −7.36655446060614346946881406783, −6.18205371221264052512287134121, −5.20581549300400466359991490832, −4.60657500121310258618406628634, −3.48119750726470909916394202472, −2.04124522009189983557643337644,
0.33900283762510085792816425473, 1.83976765163521167928633654641, 2.89482614234160538226307188203, 4.01447318220950203180659533846, 5.15246677623423518985594486824, 5.94588866048212482367291290365, 7.36968635601476475027561036474, 7.77612921283094186400315734487, 8.660162121413821916064949680429, 9.876017528067997263728010117759
