| L(s) = 1 | + (0.595 + 1.16i)2-s + (−0.315 − 1.70i)3-s + (0.163 − 0.225i)4-s + (0.0620 + 0.391i)5-s + (1.80 − 1.38i)6-s + (0.908 − 0.775i)7-s + (2.95 + 0.467i)8-s + (−2.80 + 1.07i)9-s + (−0.420 + 0.305i)10-s + (1.43 + 0.881i)11-s + (−0.435 − 0.207i)12-s + (−1.10 + 0.0866i)13-s + (1.44 + 0.599i)14-s + (0.647 − 0.229i)15-s + (1.03 + 3.20i)16-s + (−7.17 + 1.72i)17-s + ⋯ |
| L(s) = 1 | + (0.421 + 0.826i)2-s + (−0.182 − 0.983i)3-s + (0.0818 − 0.112i)4-s + (0.0277 + 0.175i)5-s + (0.736 − 0.564i)6-s + (0.343 − 0.293i)7-s + (1.04 + 0.165i)8-s + (−0.933 + 0.358i)9-s + (−0.133 + 0.0967i)10-s + (0.433 + 0.265i)11-s + (−0.125 − 0.0599i)12-s + (−0.305 + 0.0240i)13-s + (0.387 + 0.160i)14-s + (0.167 − 0.0591i)15-s + (0.259 + 0.800i)16-s + (−1.73 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33027 + 0.0306353i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33027 + 0.0306353i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.315 + 1.70i)T \) |
| 41 | \( 1 + (-4.37 + 4.67i)T \) |
| good | 2 | \( 1 + (-0.595 - 1.16i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.0620 - 0.391i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-0.908 + 0.775i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-1.43 - 0.881i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (1.10 - 0.0866i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (7.17 - 1.72i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (2.37 + 0.186i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (1.46 - 4.51i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.31 + 1.03i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-0.811 - 1.11i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.82 + 2.05i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-5.92 + 3.01i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (4.59 - 5.38i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-1.81 + 7.56i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-5.24 - 1.70i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.35 - 6.57i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (4.71 - 2.88i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-1.06 + 1.73i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (5.55 + 5.55i)T + 73iT^{2} \) |
| 79 | \( 1 + (-15.0 + 6.24i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 9.90iT - 83T^{2} \) |
| 89 | \( 1 + (8.68 + 10.1i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (0.233 + 0.380i)T + (-44.0 + 86.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.58630882275301315221421760662, −12.71138596553930089464665264904, −11.38869205923727276918308654213, −10.62774440138494371532069805578, −8.858231081150262171846660085180, −7.56255059936939613998835420966, −6.81713909235134108837604139437, −5.87922930569884096563717650576, −4.48973578464492183790036802737, −1.95657884135149068764214360281,
2.52238497182473547968109506675, 4.04809893269998136597296822158, 4.95666750733645809302777160902, 6.63530078579056974090345725611, 8.378105980554817691869094063079, 9.380574544703867423996244138408, 10.71679113359890028095526668104, 11.24958587554156430541070493603, 12.20544606772601287474914489767, 13.23656476422339782526144652606
