L(s) = 1 | + (1.79 + 1.32i)5-s + 0.476i·7-s + 3.39·11-s − 2.28i·13-s + 2.61i·17-s − 5.08·19-s − i·23-s + (1.47 + 4.77i)25-s + 3.19·29-s + 3.65·31-s + (−0.632 + 0.857i)35-s + 8.44i·37-s + 5.25·41-s + 0.269i·43-s + 8.26i·47-s + ⋯ |
L(s) = 1 | + (0.804 + 0.593i)5-s + 0.180i·7-s + 1.02·11-s − 0.634i·13-s + 0.634i·17-s − 1.16·19-s − 0.208i·23-s + (0.295 + 0.955i)25-s + 0.592·29-s + 0.655·31-s + (−0.106 + 0.145i)35-s + 1.38i·37-s + 0.820·41-s + 0.0410i·43-s + 1.20i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.342134458\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.342134458\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.79 - 1.32i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 - 0.476iT - 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 + 2.28iT - 13T^{2} \) |
| 17 | \( 1 - 2.61iT - 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 - 8.44iT - 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 - 0.269iT - 43T^{2} \) |
| 47 | \( 1 - 8.26iT - 47T^{2} \) |
| 53 | \( 1 + 7.77iT - 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 - 5.45T + 61T^{2} \) |
| 67 | \( 1 + 6.83iT - 67T^{2} \) |
| 71 | \( 1 - 6.28T + 71T^{2} \) |
| 73 | \( 1 - 4.90iT - 73T^{2} \) |
| 79 | \( 1 - 9.29T + 79T^{2} \) |
| 83 | \( 1 - 6.19iT - 83T^{2} \) |
| 89 | \( 1 - 0.423T + 89T^{2} \) |
| 97 | \( 1 - 8.67iT - 97T^{2} \) |
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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
−8.498223131228870340926444103979, −7.916921067845246343811420142164, −6.74469798420640286047047674773, −6.42677940332564067759819492593, −5.75930846047598638346983342320, −4.78336377835813125113890127717, −3.92239414939233834870044169541, −2.97873843206486480271709557734, −2.16310327698913289996960232989, −1.12922781331145911543954478605,
0.74394207488774772272624346599, 1.79208822442681414770346828480, 2.60572780682595330732985825177, 3.94354388534606140166336896779, 4.45325967082583426572503742167, 5.35876850039820174270823146669, 6.18768553666958412414303165675, 6.69982796830683601701171707300, 7.52716864531599174067812951120, 8.555684850474281133186363566149