L(s) = 1 | + i·3-s + i·7-s + 2·9-s + 11-s + 6i·13-s + 7i·17-s + 19-s − 21-s − 8i·23-s + 5i·27-s + 6·29-s − 4·31-s + i·33-s − 8i·37-s − 6·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377i·7-s + 0.666·9-s + 0.301·11-s + 1.66i·13-s + 1.69i·17-s + 0.229·19-s − 0.218·21-s − 1.66i·23-s + 0.962i·27-s + 1.11·29-s − 0.718·31-s + 0.174i·33-s − 1.31i·37-s − 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.777746041\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.777746041\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 15iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.970158244932092815366968472816, −8.578695985536066704380398280713, −7.50503683514758368259561768628, −6.61249687369567945683998830976, −6.15893917625170394440255756712, −4.96522605835732322251727207776, −4.25227263216853171430355368735, −3.72994579584096899222154700627, −2.34967194082499617238994524512, −1.43959317848638518595918552240,
0.60399767384949736125791487850, 1.54493464482122842816710214278, 2.87453571293319934561402060811, 3.57536110721281645373913201876, 4.80289243422002352876560823685, 5.38401529372002869627730417337, 6.41984218277377415154560855052, 7.21302791844666310032264327376, 7.61664346359690389974359169425, 8.381849559395472377295807458795