| L(s) = 1 | + (1.17 − 0.788i)2-s + (0.00323 + 1.73i)3-s + (0.755 − 1.85i)4-s + (0.951 + 1.42i)5-s + (1.36 + 2.03i)6-s + (−0.304 + 0.735i)7-s + (−0.573 − 2.76i)8-s + (−2.99 + 0.0112i)9-s + (2.23 + 0.920i)10-s + (5.27 + 1.05i)11-s + (3.20 + 1.30i)12-s + (−2.38 − 1.59i)13-s + (0.222 + 1.10i)14-s + (−2.46 + 1.65i)15-s + (−2.85 − 2.79i)16-s + (−3.60 − 3.60i)17-s + ⋯ |
| L(s) = 1 | + (0.830 − 0.557i)2-s + (0.00186 + 0.999i)3-s + (0.377 − 0.925i)4-s + (0.425 + 0.636i)5-s + (0.559 + 0.828i)6-s + (−0.115 + 0.278i)7-s + (−0.202 − 0.979i)8-s + (−0.999 + 0.00373i)9-s + (0.708 + 0.291i)10-s + (1.59 + 0.316i)11-s + (0.926 + 0.376i)12-s + (−0.661 − 0.441i)13-s + (0.0594 + 0.295i)14-s + (−0.635 + 0.426i)15-s + (−0.714 − 0.699i)16-s + (−0.874 − 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85897 + 0.107650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85897 + 0.107650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 + 0.788i)T \) |
| 3 | \( 1 + (-0.00323 - 1.73i)T \) |
| good | 5 | \( 1 + (-0.951 - 1.42i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.304 - 0.735i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-5.27 - 1.05i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (2.38 + 1.59i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (3.60 + 3.60i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.23 + 1.49i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-2.89 - 6.98i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.806 + 4.05i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + (2.51 + 3.76i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.08 - 0.450i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-1.84 - 0.367i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (4.93 - 4.93i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.88 - 9.47i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-2.08 + 1.39i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (0.211 + 1.06i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-13.8 + 2.75i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (-4.58 - 1.89i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.638 + 0.264i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.62 + 2.62i)T - 79iT^{2} \) |
| 83 | \( 1 + (1.68 - 2.52i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-14.6 - 6.06i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 0.149iT - 97T^{2} \) |
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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
−12.42548498477574139448963099387, −11.43404466556027979450361329221, −10.82785363513467478328996107558, −9.583922444462409398493564112967, −9.242019466257142653220576225649, −7.02900453963783056191961468006, −5.98994771279611925951912916624, −4.83540573198232836879426959922, −3.69697982464186402697686875307, −2.43766272881993116446299771466,
1.91201717942424375047601924604, 3.79460603042396726403381179506, 5.17015104544727648632860738160, 6.49525238296894041559657974278, 6.89672007514188955733734155504, 8.437241504228139430179292287485, 9.041275857197121474643004567373, 10.96605000289107294524338850550, 11.97181821120794469977093214694, 12.75077761729879073338105955194
