| L(s) = 1 | − i·3-s + i·5-s + 1.52·7-s − 9-s + 6.43·11-s − 4.50·13-s + 15-s + 1.65i·17-s − 5.57·19-s − 1.52i·21-s + (−2.99 + 3.74i)23-s − 25-s + i·27-s − 2.30·29-s + 1.77i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s + 0.575·7-s − 0.333·9-s + 1.93·11-s − 1.24·13-s + 0.258·15-s + 0.400i·17-s − 1.27·19-s − 0.332i·21-s + (−0.623 + 0.781i)23-s − 0.200·25-s + 0.192i·27-s − 0.427·29-s + 0.318i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028739624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028739624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 23 | \( 1 + (2.99 - 3.74i)T \) |
| good | 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 6.43T + 11T^{2} \) |
| 13 | \( 1 + 4.50T + 13T^{2} \) |
| 17 | \( 1 - 1.65iT - 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 - 1.77iT - 31T^{2} \) |
| 37 | \( 1 - 5.43iT - 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 - 4.75T + 43T^{2} \) |
| 47 | \( 1 - 2.99iT - 47T^{2} \) |
| 53 | \( 1 + 2.72iT - 53T^{2} \) |
| 59 | \( 1 - 3.99iT - 59T^{2} \) |
| 61 | \( 1 - 9.74iT - 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 - 2.39iT - 71T^{2} \) |
| 73 | \( 1 + 9.23T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.35T + 83T^{2} \) |
| 89 | \( 1 + 2.66iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−8.388486200071560452365316234084, −7.49084497741282418347250249748, −7.01693788286866481563006784320, −6.30662940239585806714853273392, −5.72142375693703797901203127557, −4.56832291772617776728643418060, −4.03778841583843789477214196859, −3.01627854175055719788151892051, −1.98010222040107113169356830421, −1.36838048369978136692325983144,
0.25410489186048062377314624389, 1.62198072962898017473813778047, 2.43276323736338279455315115698, 3.69414407749774500501982276050, 4.35043183366226278875430301358, 4.77171940908008937815221104001, 5.75583687602217333138210449352, 6.46887998434609671576367061334, 7.20833768232960957197666859403, 8.054814624066879171998053078840
