L(s) = 1 | − 3i·3-s + 2i·7-s − 6·9-s + 11-s − 6i·17-s + 4·19-s + 6·21-s + i·23-s + 9i·27-s + 8·29-s + 7·31-s − 3i·33-s − i·37-s + 4·41-s + 6i·43-s + ⋯ |
L(s) = 1 | − 1.73i·3-s + 0.755i·7-s − 2·9-s + 0.301·11-s − 1.45i·17-s + 0.917·19-s + 1.30·21-s + 0.208i·23-s + 1.73i·27-s + 1.48·29-s + 1.25·31-s − 0.522i·33-s − 0.164i·37-s + 0.624·41-s + 0.914i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.901789022\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.901789022\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−7.951431383324858364901210758269, −7.41209920928680676984447504326, −6.70388946021883691305078858621, −6.12672976890198425998295788291, −5.39381535128777363073752519237, −4.53001543068568994012316134246, −2.91708226982043547549471154008, −2.69487002432059801616202550248, −1.48746709163632682711682285924, −0.67020504408855193567243288286,
1.00102072246229447522872074416, 2.57937755004782498892208386134, 3.54331587082993759977051842019, 4.05569850635683108654609145824, 4.69262992226800341002375474446, 5.47895167395053194509541917472, 6.26457842295921585138265576609, 7.09981035968807620092504832390, 8.210438531955501446103592766434, 8.591158913313704180525562879618