| L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.03 − 0.598i)5-s + (2.52 + 0.782i)7-s + 0.999·8-s + (−1.03 − 0.598i)10-s + (−2.65 + 1.98i)11-s + 3.26i·13-s + (−0.586 − 2.57i)14-s + (−0.5 − 0.866i)16-s + (2.21 − 3.82i)17-s + (6.73 − 3.88i)19-s + 1.19i·20-s + (3.04 + 1.30i)22-s + (−5.94 + 3.43i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.463 − 0.267i)5-s + (0.955 + 0.295i)7-s + 0.353·8-s + (−0.327 − 0.189i)10-s + (−0.800 + 0.599i)11-s + 0.904i·13-s + (−0.156 − 0.689i)14-s + (−0.125 − 0.216i)16-s + (0.536 − 0.928i)17-s + (1.54 − 0.892i)19-s + 0.267i·20-s + (0.650 + 0.277i)22-s + (−1.24 + 0.715i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613373526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613373526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.52 - 0.782i)T \) |
| 11 | \( 1 + (2.65 - 1.98i)T \) |
| good | 5 | \( 1 + (-1.03 + 0.598i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 3.26iT - 13T^{2} \) |
| 17 | \( 1 + (-2.21 + 3.82i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.73 + 3.88i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.94 - 3.43i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + (0.103 - 0.179i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.83 - 6.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.15T + 41T^{2} \) |
| 43 | \( 1 - 8.00iT - 43T^{2} \) |
| 47 | \( 1 + (-0.355 + 0.205i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.02 + 2.90i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.66 + 2.11i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.94 - 5.74i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.61 + 9.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.80iT - 71T^{2} \) |
| 73 | \( 1 + (-0.195 - 0.113i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.00 + 1.15i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + (-11.4 + 6.58i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−9.557981189816515221434497759743, −9.034730552111588945404275698793, −7.80118404558286369185399387349, −7.58281836256804092440792452740, −6.21506858685592424935986710691, −4.99692409042463038113774899437, −4.73448060445760164325093326328, −3.18848777385339971980191450789, −2.19825174328447752609508069237, −1.20940014351285856381503689696,
0.889117499584959341963286712742, 2.25167460977169578300517336908, 3.55939786777675373112761280324, 4.74539564978041676895471094877, 5.73920122325092693360431602781, 6.05023101592860796940240908804, 7.45346538247448971061399270033, 7.996216375461652034084185149980, 8.436212554339172280666149260481, 9.696715616356575896266733216163
