| L(s) = 1 | + (0.5 − 1.65i)3-s + 2·4-s + (−1.5 − 1.65i)5-s + (−2.5 − 1.65i)9-s + 3.31i·11-s + (1 − 3.31i)12-s + (−3.5 + 1.65i)15-s + 4·16-s + (−3 − 3.31i)20-s + 9·23-s + (−0.5 + 4.97i)25-s + (−4 + 3.31i)27-s − 5·31-s + (5.5 + 1.65i)33-s + (−5 − 3.31i)36-s + 9.94i·37-s + ⋯ |
| L(s) = 1 | + (0.288 − 0.957i)3-s + 4-s + (−0.670 − 0.741i)5-s + (−0.833 − 0.552i)9-s + 1.00i·11-s + (0.288 − 0.957i)12-s + (−0.903 + 0.428i)15-s + 16-s + (−0.670 − 0.741i)20-s + 1.87·23-s + (−0.100 + 0.994i)25-s + (−0.769 + 0.638i)27-s − 0.898·31-s + (0.957 + 0.288i)33-s + (−0.833 − 0.552i)36-s + 1.63i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13230 - 0.716483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13230 - 0.716483i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.5 + 1.65i)T \) |
| 5 | \( 1 + (1.5 + 1.65i)T \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 9.94iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 9.94iT - 67T^{2} \) |
| 71 | \( 1 + 16.5iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 9.94iT - 97T^{2} \) |
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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
−12.57789636297537256031397209629, −11.84191786604134562481887679611, −11.00393870103648858722226618268, −9.450085728632858472179469831778, −8.267083628327105163353433029816, −7.38352806560234754153978259748, −6.60200565249876502692010927368, −5.04461147530369408104381791515, −3.17753460474906485479382127526, −1.55857966678580133403849004960,
2.79401446676268275688596886172, 3.69871258988377707656808652426, 5.42814016399596099780977689678, 6.73579481187108833581392021092, 7.82606779128768332906590611201, 8.925913199810205903805379265754, 10.28452879113529251011115213399, 11.12901341358793320730018418003, 11.46194658011896096540343981811, 12.97981044767643570834059958272
