| L(s) = 1 | + i·2-s − 4-s + 1.06·5-s + (0.884 − 2.49i)7-s − i·8-s + 1.06i·10-s − i·11-s + (2.49 + 0.884i)14-s + 16-s − 3.91·17-s − 7.49i·19-s − 1.06·20-s + 22-s − 5.33i·23-s − 3.85·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.478·5-s + (0.334 − 0.942i)7-s − 0.353i·8-s + 0.338i·10-s − 0.301i·11-s + (0.666 + 0.236i)14-s + 0.250·16-s − 0.950·17-s − 1.71i·19-s − 0.239·20-s + 0.213·22-s − 1.11i·23-s − 0.771·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323360758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323360758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.884 + 2.49i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.06T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3.91T + 17T^{2} \) |
| 19 | \( 1 + 7.49iT - 19T^{2} \) |
| 23 | \( 1 + 5.33iT - 23T^{2} \) |
| 29 | \( 1 - 8.01iT - 29T^{2} \) |
| 31 | \( 1 + 3.65iT - 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 0.230T + 43T^{2} \) |
| 47 | \( 1 - 0.370T + 47T^{2} \) |
| 53 | \( 1 + 7.56iT - 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 + 14.2iT - 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 3.15iT - 71T^{2} \) |
| 73 | \( 1 + 0.370iT - 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−9.227095633486084599363835726322, −8.679931861478424627075176136870, −7.74407365933552302117543631410, −6.86554754355170227531700767728, −6.43255805095231727219498986477, −5.18084886733587794468334161416, −4.59672317798294441389997814066, −3.52432593760422080007073162005, −2.12297871926807049308506228298, −0.52714240050751429611538598692,
1.63989062515641759359024506570, 2.31524535992871579662875549010, 3.55648454744687246183839416054, 4.53520216030556369312090946442, 5.59949618541280871314652247888, 6.08566402623240550877091369271, 7.43514007282953808393834097139, 8.284535070344474896527763511591, 9.040276395903837191319564266596, 9.757097057315385438297205699720
