| L(s) = 1 | + i·3-s + (−0.847 − 2.06i)5-s − 3.67i·7-s − 9-s − 3.98·11-s + 1.24i·13-s + (2.06 − 0.847i)15-s + 0.354i·17-s − 1.69·19-s + 3.67·21-s + 1.36i·23-s + (−3.56 + 3.50i)25-s − i·27-s + 0.677·29-s + 0.344·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.379 − 0.925i)5-s − 1.38i·7-s − 0.333·9-s − 1.20·11-s + 0.346i·13-s + (0.534 − 0.218i)15-s + 0.0858i·17-s − 0.387·19-s + 0.801·21-s + 0.284i·23-s + (−0.712 + 0.701i)25-s − 0.192i·27-s + 0.125·29-s + 0.0618·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3323019677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3323019677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.847 + 2.06i)T \) |
| 67 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.67iT - 7T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 - 1.24iT - 13T^{2} \) |
| 17 | \( 1 - 0.354iT - 17T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 - 1.36iT - 23T^{2} \) |
| 29 | \( 1 - 0.677T + 29T^{2} \) |
| 31 | \( 1 - 0.344T + 31T^{2} \) |
| 37 | \( 1 + 8.16iT - 37T^{2} \) |
| 41 | \( 1 + 5.45T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 10.1iT - 47T^{2} \) |
| 53 | \( 1 + 4.74iT - 53T^{2} \) |
| 59 | \( 1 - 5.13T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 - 7.49iT - 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 17.6iT - 97T^{2} \) |
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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
−8.584770133538296702106958632884, −7.984367615442820482389753760828, −7.38175037094347775446030853986, −6.53301431895511255883972703384, −5.38994462922118434430296728833, −4.91501609699515467323453733715, −4.06191407199640111116251811881, −3.60285278580153468342001183701, −2.29024444027464271285965698113, −0.975545527366455951186270920974,
0.10781105551788223565013949916, 1.90556505200146854873133157275, 2.72042003759165761755195923228, 3.15407530465865373570542536036, 4.50266433654314162932632246816, 5.46125972702817438234072336723, 5.97940113051864612984352919684, 6.80092031827859918570551505598, 7.47180590948857050130842186422, 8.305565595554742800596317513832
