| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s − 3.46·7-s + 2.82·8-s + 2.82i·11-s − 3.46·13-s + (2.44 − 4.24i)14-s + (−2.00 + 3.46i)16-s − 1.41·17-s + 4·19-s + (−3.46 − 2.00i)22-s − 4.89i·23-s + (2.44 − 4.24i)26-s + (3.46 + 5.99i)28-s + 2.44·29-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.30·7-s + 0.999·8-s + 0.852i·11-s − 0.960·13-s + (0.654 − 1.13i)14-s + (−0.500 + 0.866i)16-s − 0.342·17-s + 0.917·19-s + (−0.738 − 0.426i)22-s − 1.02i·23-s + (0.480 − 0.832i)26-s + (0.654 + 1.13i)28-s + 0.454·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7012903608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7012903608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4.89iT - 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4.89iT - 47T^{2} \) |
| 53 | \( 1 + 7.34iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.415525616747232028306807323649, −8.423072721104098039198789251716, −7.58458224654718410040429094517, −6.76297645774327552227920302931, −6.44804717057404132029712848731, −5.23329031492121830695238940917, −4.60692354705453415972290183650, −3.34972925825602436764859572617, −2.11282568867833166916981167291, −0.41708980443989370355549894997,
0.869458947608028134945088343945, 2.44027755652148407801509242181, 3.18928527808570787138967990132, 3.96192799954530771303165357782, 5.14671637370524923775865069069, 6.11544008355345671630368749852, 7.15650961684813165295992555459, 7.75310478020643204066045598551, 8.841020359409892236208331441391, 9.391805172692858142869848112415
