L(s) = 1 | − i·7-s + 6·11-s + 5i·13-s − 6i·17-s + 5·19-s − 6i·23-s − 6·29-s + 31-s − 2i·37-s + i·43-s − 6i·47-s + 6·49-s + 12i·53-s + 6·59-s − 13·61-s + ⋯ |
L(s) = 1 | − 0.377i·7-s + 1.80·11-s + 1.38i·13-s − 1.45i·17-s + 1.14·19-s − 1.25i·23-s − 1.11·29-s + 0.179·31-s − 0.328i·37-s + 0.152i·43-s − 0.875i·47-s + 0.857·49-s + 1.64i·53-s + 0.781·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.167585746\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167585746\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.771812430891067718861478482068, −7.51602132357373174389843400926, −7.01890935575847896116117761371, −6.45244928990761226475218055610, −5.50584088868085216328610878521, −4.45307051704416614190699098179, −4.02582808433159564164356811089, −2.97972014926175197134583432130, −1.82065939696385397107901668761, −0.810328700324261265801925985096,
1.04349612918029643659575131225, 1.92683348019114953411336751075, 3.40283852643378920439273605297, 3.65673988935779684183758624800, 4.87370293754579118656211810960, 5.78113002357881209574159147094, 6.17726692128178121108605732325, 7.21275665529451835934570381372, 7.86179720746328266290098440204, 8.629679674318943130088201720546