| L(s) = 1 | + 1.56i·3-s + i·7-s + 0.561·9-s + 1.56·11-s − 0.438i·13-s − 0.438i·17-s − 7.12·19-s − 1.56·21-s + 3.12i·23-s + 5.56i·27-s − 6.68·29-s + 2.43i·33-s + 6i·37-s + 0.684·39-s + 5.12·41-s + ⋯ |
| L(s) = 1 | + 0.901i·3-s + 0.377i·7-s + 0.187·9-s + 0.470·11-s − 0.121i·13-s − 0.106i·17-s − 1.63·19-s − 0.340·21-s + 0.651i·23-s + 1.07i·27-s − 1.24·29-s + 0.424i·33-s + 0.986i·37-s + 0.109·39-s + 0.800·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.263521550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.263521550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 1.56iT - 3T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 0.438iT - 13T^{2} \) |
| 17 | \( 1 + 0.438iT - 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 - 0.876iT - 43T^{2} \) |
| 47 | \( 1 - 8.68iT - 47T^{2} \) |
| 53 | \( 1 - 5.12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 1.12T + 89T^{2} \) |
| 97 | \( 1 - 5.80iT - 97T^{2} \) |
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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
−9.281815939424230875719285939011, −8.523851688002428944185853336417, −7.68534557230671574830574214640, −6.77894734661240783235081473103, −5.99801781294293740747761391398, −5.16225057799263856675908291233, −4.30388525667918335338218214118, −3.75713174986157639178994331638, −2.64221845860577361641568807484, −1.48428517972698366486251587448,
0.39792314554983039333910740301, 1.66556931333846498129795453625, 2.38940677915063738726660627051, 3.82919342115948216994089682906, 4.34551266295589190708727160942, 5.55622158556878941538398932387, 6.42087145397753030102590998480, 6.93192376603501862357408550291, 7.61480079074203643030036007921, 8.417970029565876307586477310058
