L(s) = 1 | + (0.258 − 2.22i)5-s − 2.14i·7-s − 4.93·11-s − 0.809i·13-s − 6.88i·17-s + 4.52·19-s + i·23-s + (−4.86 − 1.14i)25-s + 4.44·29-s + 4.11·31-s + (−4.76 − 0.554i)35-s − 4.35i·37-s − 10.6·41-s + 9.56i·43-s − 8.41i·47-s + ⋯ |
L(s) = 1 | + (0.115 − 0.993i)5-s − 0.811i·7-s − 1.48·11-s − 0.224i·13-s − 1.66i·17-s + 1.03·19-s + 0.208i·23-s + (−0.973 − 0.229i)25-s + 0.824·29-s + 0.739·31-s + (−0.805 − 0.0936i)35-s − 0.716i·37-s − 1.65·41-s + 1.45i·43-s − 1.22i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9500628494\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9500628494\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.258 + 2.22i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 2.14iT - 7T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 13 | \( 1 + 0.809iT - 13T^{2} \) |
| 17 | \( 1 + 6.88iT - 17T^{2} \) |
| 19 | \( 1 - 4.52T + 19T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 4.11T + 31T^{2} \) |
| 37 | \( 1 + 4.35iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 9.56iT - 43T^{2} \) |
| 47 | \( 1 + 8.41iT - 47T^{2} \) |
| 53 | \( 1 + 3.13iT - 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 + 9.98T + 61T^{2} \) |
| 67 | \( 1 + 0.0193iT - 67T^{2} \) |
| 71 | \( 1 + 0.834T + 71T^{2} \) |
| 73 | \( 1 - 1.48iT - 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 15.3iT - 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.935511098429645554641083648540, −7.49253275857559824931106083381, −6.69404692561137057940382461236, −5.50366926034523211458854575956, −5.08909688014727727056277076076, −4.45787886514334853742217838312, −3.30923313541352462732231412084, −2.51734995818325833288038962199, −1.17171639428241073987595647831, −0.27968631915747490171003937512,
1.64501203824917146627390963668, 2.64435832141599795820320811058, 3.13397987398991981175828570953, 4.23207603879184775516342560125, 5.28361199939098453988413854344, 5.82169371774644885715386386214, 6.57353713529648136350260199728, 7.32614750258780941209506435220, 8.178263994865957835779060994936, 8.539837998851230090064138000630