L(s) = 1 | + (−1.60 − 0.649i)3-s + (−0.468 − 0.811i)5-s + (−0.5 + 0.866i)7-s + (2.15 + 2.08i)9-s + (−2.48 + 4.30i)11-s + (−0.622 − 1.07i)13-s + (0.225 + 1.60i)15-s + 5.22·17-s + 5.18·19-s + (1.36 − 1.06i)21-s + (1.00 + 1.73i)23-s + (2.06 − 3.57i)25-s + (−2.10 − 4.74i)27-s + (−3.43 + 5.95i)29-s + (2.86 + 4.96i)31-s + ⋯ |
L(s) = 1 | + (−0.927 − 0.374i)3-s + (−0.209 − 0.362i)5-s + (−0.188 + 0.327i)7-s + (0.718 + 0.695i)9-s + (−0.749 + 1.29i)11-s + (−0.172 − 0.298i)13-s + (0.0581 + 0.414i)15-s + 1.26·17-s + 1.18·19-s + (0.297 − 0.232i)21-s + (0.209 + 0.362i)23-s + (0.412 − 0.714i)25-s + (−0.405 − 0.913i)27-s + (−0.638 + 1.10i)29-s + (0.514 + 0.891i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.864332 + 0.280496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.864332 + 0.280496i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.60 + 0.649i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.468 + 0.811i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.48 - 4.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.622 + 1.07i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 5.18T + 19T^{2} \) |
| 23 | \( 1 + (-1.00 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.43 - 5.95i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.86 - 4.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.73T + 37T^{2} \) |
| 41 | \( 1 + (-5.73 - 9.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.80 - 8.31i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.984 + 1.70i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.63T + 53T^{2} \) |
| 59 | \( 1 + (2.43 + 4.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 2.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.573 - 0.994i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + (-6.05 + 10.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.431 + 0.747i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (3.78 - 6.55i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.11997856137327640687295841088, −10.06385262268491455063988869043, −9.543428556774815965018260087600, −7.957376836209568494920106751855, −7.48437077644404838829719437718, −6.35192622700536072449739437061, −5.25665507590211623045680511381, −4.70919705391989326264456501014, −2.95453699176211040761489115804, −1.28060752794301789759996441900,
0.71719343507070073935222794465, 3.03041912759893241808004145792, 4.04372120918891461373073898039, 5.39524394582649719854872162400, 5.95816277829176321517181297383, 7.18813561378068043131444912141, 7.932809062485240784086995649381, 9.335398170746424548974934128302, 10.06210197885271285757823444099, 10.96062514816282333194666979711