| L(s) = 1 | − 2.19i·3-s + (1.64 − 1.51i)5-s + i·7-s − 1.83·9-s − 1.37·11-s + 2.74i·13-s + (−3.32 − 3.61i)15-s − 6.94i·17-s − 1.29·19-s + 2.19·21-s − 8.31i·23-s + (0.412 − 4.98i)25-s − 2.56i·27-s + 8.40·29-s − 9.49·31-s + ⋯ |
| L(s) = 1 | − 1.26i·3-s + (0.735 − 0.677i)5-s + 0.377i·7-s − 0.610·9-s − 0.414·11-s + 0.760i·13-s + (−0.859 − 0.933i)15-s − 1.68i·17-s − 0.296·19-s + 0.479·21-s − 1.73i·23-s + (0.0825 − 0.996i)25-s − 0.494i·27-s + 1.56·29-s − 1.70·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611478081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611478081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.64 + 1.51i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 2.19iT - 3T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2.74iT - 13T^{2} \) |
| 17 | \( 1 + 6.94iT - 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 + 8.31iT - 23T^{2} \) |
| 29 | \( 1 - 8.40T + 29T^{2} \) |
| 31 | \( 1 + 9.49T + 31T^{2} \) |
| 37 | \( 1 + 1.73iT - 37T^{2} \) |
| 41 | \( 1 + 5.30T + 41T^{2} \) |
| 43 | \( 1 - 7.83iT - 43T^{2} \) |
| 47 | \( 1 + 3.48iT - 47T^{2} \) |
| 53 | \( 1 - 6.13iT - 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 6.59T + 61T^{2} \) |
| 67 | \( 1 + 1.66iT - 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 - 2.13iT - 73T^{2} \) |
| 79 | \( 1 - 5.45T + 79T^{2} \) |
| 83 | \( 1 - 2.54iT - 83T^{2} \) |
| 89 | \( 1 + 1.43T + 89T^{2} \) |
| 97 | \( 1 - 9.69iT - 97T^{2} \) |
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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
−9.342063918001430599838127448037, −8.672405552679703914151064357681, −7.87542219422030990317117328777, −6.84882375199903376627807614150, −6.39007345461125351452102554681, −5.28158059617037574602734546512, −4.52715536072389343182417756830, −2.69497627617585194028274129723, −1.97705636880489886589171488896, −0.69952358444529258443580986823,
1.80458701356395642858510075695, 3.26989393520487702280468685592, 3.82698563331389767977512243193, 5.08869663698675835505780575385, 5.70823521496195519664235029503, 6.71333165339048538161330033256, 7.72962476325490237343441677090, 8.675250926318327928869972303642, 9.611897094794763487684317714646, 10.24181196204751906784474950279
