| L(s) = 1 | − i·3-s + (2.21 + 0.311i)5-s − 2.21i·7-s − 9-s − 1.31·11-s − 0.836i·13-s + (0.311 − 2.21i)15-s − 4.90i·17-s − 19-s − 2.21·21-s − 5.33i·23-s + (4.80 + 1.37i)25-s + i·27-s − 1.31·29-s − 1.71·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.990 + 0.139i)5-s − 0.836i·7-s − 0.333·9-s − 0.395·11-s − 0.232i·13-s + (0.0803 − 0.571i)15-s − 1.18i·17-s − 0.229·19-s − 0.483·21-s − 1.11i·23-s + (0.961 + 0.275i)25-s + 0.192i·27-s − 0.243·29-s − 0.308·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.695158947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695158947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.21 - 0.311i)T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2.21iT - 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 + 0.836iT - 13T^{2} \) |
| 17 | \( 1 + 4.90iT - 17T^{2} \) |
| 23 | \( 1 + 5.33iT - 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 + 1.03iT - 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 + 0.214iT - 43T^{2} \) |
| 47 | \( 1 + 5.33iT - 47T^{2} \) |
| 53 | \( 1 - 8.70iT - 53T^{2} \) |
| 59 | \( 1 + 8.10T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 + 9.61iT - 67T^{2} \) |
| 71 | \( 1 + 7.80T + 71T^{2} \) |
| 73 | \( 1 - 0.193iT - 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 0.0459iT - 83T^{2} \) |
| 89 | \( 1 - 2.92T + 89T^{2} \) |
| 97 | \( 1 + 2.60iT - 97T^{2} \) |
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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
−9.579799429429563843051956948537, −8.847846911193533643494801704851, −7.75937091747736543025049954800, −7.09558884497957093356959411633, −6.30968262397570344003299826464, −5.42226312320264516231707164490, −4.46816613434018754215829761247, −3.05497657689880742790799365731, −2.12631196295691755050445993095, −0.73489873080839421056084159338,
1.70426795010880710693524778281, 2.71355682412500713932923336799, 3.91205448948126330924139248018, 5.06841236641930070135690589532, 5.73868525732265095240782470099, 6.39489750410319123299217070072, 7.69076158395218465740855381013, 8.660271900691801114737307818520, 9.239495730012006811121550347996, 9.958172295982792188276549084025
