| L(s) = 1 | − 3.13i·3-s + i·5-s + 7-s − 6.82·9-s + 4.07i·11-s − 2.86i·13-s + 3.13·15-s + 7.69·17-s + 2.79i·19-s − 3.13i·21-s + 7.79·23-s − 25-s + 11.9i·27-s − 9.95i·29-s − 0.232·31-s + ⋯ |
| L(s) = 1 | − 1.80i·3-s + 0.447i·5-s + 0.377·7-s − 2.27·9-s + 1.22i·11-s − 0.795i·13-s + 0.809·15-s + 1.86·17-s + 0.640i·19-s − 0.684i·21-s + 1.62·23-s − 0.200·25-s + 2.30i·27-s − 1.84i·29-s − 0.0417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.836574472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836574472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.13iT - 3T^{2} \) |
| 11 | \( 1 - 4.07iT - 11T^{2} \) |
| 13 | \( 1 + 2.86iT - 13T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 - 2.79iT - 19T^{2} \) |
| 23 | \( 1 - 7.79T + 23T^{2} \) |
| 29 | \( 1 + 9.95iT - 29T^{2} \) |
| 31 | \( 1 + 0.232T + 31T^{2} \) |
| 37 | \( 1 + 4.98iT - 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 + 9.54iT - 43T^{2} \) |
| 47 | \( 1 + 3.90T + 47T^{2} \) |
| 53 | \( 1 + 3.85iT - 53T^{2} \) |
| 59 | \( 1 - 7.86iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 11.5iT - 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 5.99T + 73T^{2} \) |
| 79 | \( 1 - 0.872T + 79T^{2} \) |
| 83 | \( 1 + 8.33iT - 83T^{2} \) |
| 89 | \( 1 + 5.79T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.409520400709277918340739030611, −7.80011370642271301751565442186, −7.36223406070665252664709086561, −6.70067527348242009623242022310, −5.75696509322654665807308651066, −5.17350899291431752229327526930, −3.63057507572802121546898733285, −2.63435510656901456696754924529, −1.79476038464155758705570997704, −0.76835561935458413373903923913,
1.13038714543321346095461811374, 3.09648090344800286546272995507, 3.41467657667881575867086365613, 4.61532394022890635154861246454, 5.09763800606788475163127969209, 5.73113562620443656243381608484, 6.85625769964816355972205274137, 8.127456902068610770384042120629, 8.656667536016744072801764401448, 9.344037488226250818143689569443
