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
−8.305565595554742800596317513832, −7.47180590948857050130842186422, −6.80092031827859918570551505598, −5.97940113051864612984352919684, −5.46125972702817438234072336723, −4.50266433654314162932632246816, −3.15407530465865373570542536036, −2.72042003759165761755195923228, −1.90556505200146854873133157275, −0.10781105551788223565013949916,
0.975545527366455951186270920974, 2.29024444027464271285965698113, 3.60285278580153468342001183701, 4.06191407199640111116251811881, 4.91501609699515467323453733715, 5.38994462922118434430296728833, 6.53301431895511255883972703384, 7.38175037094347775446030853986, 7.984367615442820482389753760828, 8.584770133538296702106958632884