| L(s) = 1 | + i·3-s − 2.82i·5-s − 9-s − i·11-s + 6.29i·13-s + 2.82·15-s − 6.89·17-s − 4.89i·19-s − 6.29·23-s − 3.00·25-s − i·27-s + 0.635i·29-s + 9.12·31-s + 33-s + 6.92i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.26i·5-s − 0.333·9-s − 0.301i·11-s + 1.74i·13-s + 0.730·15-s − 1.67·17-s − 1.12i·19-s − 1.31·23-s − 0.600·25-s − 0.192i·27-s + 0.118i·29-s + 1.63·31-s + 0.174·33-s + 1.13i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1688209533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1688209533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 6.29iT - 13T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 - 0.635iT - 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8.89iT - 43T^{2} \) |
| 47 | \( 1 + 0.635T + 47T^{2} \) |
| 53 | \( 1 - 9.75iT - 53T^{2} \) |
| 59 | \( 1 - 5.79iT - 59T^{2} \) |
| 61 | \( 1 + 5.02iT - 61T^{2} \) |
| 67 | \( 1 - 13.7iT - 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 9.79iT - 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.326813512187420402836316405846, −8.637221984964151112159349590495, −8.400423883547616911253701232953, −6.92741424539998343742992322293, −6.38579109066556856358493103585, −5.23380504741661233277671563413, −4.44862827878681488627882698899, −4.17985061060592904761965128136, −2.64581290731530946762623453146, −1.53147994610171845358541093915,
0.05573700954874060384421435263, 1.84477891261726450563088471071, 2.75784816381058023171367726900, 3.53343119841209111722155295700, 4.70469277606990334546499683681, 5.88510433995240456063758747858, 6.40352813267813197522079982362, 7.12921632154172679376138044428, 8.033400018496237846793990822170, 8.359355438276217420122654845213
