L(s) = 1 | − i·3-s − 1.41i·5-s − 4.24·7-s − 9-s − 6i·11-s + 5.65i·13-s − 1.41·15-s − 6·17-s + 4i·19-s + 4.24i·21-s + 2.82·23-s + 2.99·25-s + i·27-s + 1.41i·29-s + 1.41·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.632i·5-s − 1.60·7-s − 0.333·9-s − 1.80i·11-s + 1.56i·13-s − 0.365·15-s − 1.45·17-s + 0.917i·19-s + 0.925i·21-s + 0.589·23-s + 0.599·25-s + 0.192i·27-s + 0.262i·29-s + 0.254·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3516452238\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3516452238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.332233705367633635075697254626, −8.909144255799384661855650605407, −8.286101931560156626507970093116, −6.96285332278553630232707467739, −6.44892099048159614654016620824, −5.86420289870004039399181292456, −4.59954440103965887802302073423, −3.56263225918027442201960691961, −2.69203294738754593302184304092, −1.22283087260145487921329597734,
0.14421751405532148046622919484, 2.46387766530499081724185067589, 3.05481220348225492357865504944, 4.14372872299324192402002588248, 5.01488211565020311430219594598, 6.08690711260584154577645668082, 6.92343175259227591410148638706, 7.34494066004496203472988037204, 8.717712687630152429699978437715, 9.404745961272412481153501627114