| L(s) = 1 | + 3.32i·3-s − 0.885·5-s − i·7-s − 8.06·9-s + 4.99i·11-s + 4.26i·13-s − 2.94i·15-s − 5.28i·17-s + 0.754i·19-s + 3.32·21-s − 2.59·23-s − 4.21·25-s − 16.8i·27-s − 2.99i·29-s + 2.96·31-s + ⋯ |
| L(s) = 1 | + 1.92i·3-s − 0.396·5-s − 0.377i·7-s − 2.68·9-s + 1.50i·11-s + 1.18i·13-s − 0.760i·15-s − 1.28i·17-s + 0.172i·19-s + 0.725·21-s − 0.540·23-s − 0.843·25-s − 3.24i·27-s − 0.556i·29-s + 0.533·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6394387770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6394387770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 41 | \( 1 + (5.15 + 3.80i)T \) |
| good | 3 | \( 1 - 3.32iT - 3T^{2} \) |
| 5 | \( 1 + 0.885T + 5T^{2} \) |
| 11 | \( 1 - 4.99iT - 11T^{2} \) |
| 13 | \( 1 - 4.26iT - 13T^{2} \) |
| 17 | \( 1 + 5.28iT - 17T^{2} \) |
| 19 | \( 1 - 0.754iT - 19T^{2} \) |
| 23 | \( 1 + 2.59T + 23T^{2} \) |
| 29 | \( 1 + 2.99iT - 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 2.19iT - 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 0.407T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 + 7.13iT - 67T^{2} \) |
| 71 | \( 1 + 13.4iT - 71T^{2} \) |
| 73 | \( 1 + 1.38T + 73T^{2} \) |
| 79 | \( 1 - 10.5iT - 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 7.56iT - 89T^{2} \) |
| 97 | \( 1 - 18.7iT - 97T^{2} \) |
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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
−10.17206197061209785783654654529, −9.533306800255148907867565381506, −9.143126426783859559590478064561, −7.976386197488402071703207852642, −7.08842722016557503096075276945, −5.91572000089106945672393909259, −4.71960810274263453743391005766, −4.43833275130011545734889984889, −3.58934366890530082965918360589, −2.32424996601990485290056912107,
0.27411636304824066348005433532, 1.49631630346978660675866044793, 2.73137455534944656133665343965, 3.58006722918869538983858272232, 5.49749247785796572315694929109, 5.96131000634831994576518854725, 6.74740266662627759799098053590, 7.78831787278956058433776846001, 8.292513148712229380612430946072, 8.700007606590934278415774357431
