L(s) = 1 | + (1.40 − 0.112i)2-s + (0.707 + 0.707i)3-s + (1.97 − 0.316i)4-s + (−0.466 + 2.18i)5-s + (1.07 + 0.917i)6-s − 1.00·7-s + (2.74 − 0.667i)8-s + 1.00i·9-s + (−0.413 + 3.13i)10-s + (−1.89 − 1.89i)11-s + (1.61 + 1.17i)12-s + (−2.65 − 2.65i)13-s + (−1.40 + 0.112i)14-s + (−1.87 + 1.21i)15-s + (3.80 − 1.24i)16-s + 1.73i·17-s + ⋯ |
L(s) = 1 | + (0.996 − 0.0792i)2-s + (0.408 + 0.408i)3-s + (0.987 − 0.158i)4-s + (−0.208 + 0.977i)5-s + (0.439 + 0.374i)6-s − 0.378·7-s + (0.971 − 0.235i)8-s + 0.333i·9-s + (−0.130 + 0.991i)10-s + (−0.571 − 0.571i)11-s + (0.467 + 0.338i)12-s + (−0.737 − 0.737i)13-s + (−0.376 + 0.0299i)14-s + (−0.484 + 0.314i)15-s + (0.950 − 0.312i)16-s + 0.421i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21387 + 0.565020i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21387 + 0.565020i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 + 0.112i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.466 - 2.18i)T \) |
good | 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 + (1.89 + 1.89i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.65 + 2.65i)T + 13iT^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + (-5.33 + 5.33i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.160T + 23T^{2} \) |
| 29 | \( 1 + (-2.70 + 2.70i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + (5.35 - 5.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (-7.23 + 7.23i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.79iT - 47T^{2} \) |
| 53 | \( 1 + (3.44 - 3.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.101 - 0.101i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.01 - 6.01i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.04 + 9.04i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.60iT - 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 + (2.04 + 2.04i)T + 83iT^{2} \) |
| 89 | \( 1 - 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 3.84iT - 97T^{2} \) |
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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
−12.26804002753464631514889519085, −11.21187274027475235757744931743, −10.51963599535173958687874067270, −9.620380904755338028696343009750, −7.959835555798791754890422549141, −7.13671902048191007170945538366, −5.94941532002498377097074796060, −4.80928061559200491340764855500, −3.33797140220774806870535730308, −2.71864489357790385940578767572,
1.90520791722310679332018252849, 3.46518824452045437091568028452, 4.72665672581294174822684916701, 5.68699427031778291723456899339, 7.14100488202483894935303544939, 7.76262584785059934946352110836, 9.109213181182375994542983783829, 10.12394712975701035124942617068, 11.58800986204876359383545440459, 12.41712386161085721264877440345