L(s) = 1 | + (0.111 + 0.111i)5-s − 7-s + (3.61 − 3.61i)11-s + (−1.94 − 1.94i)13-s − 4.79i·17-s + (−3.03 + 3.03i)19-s + 6.58i·23-s − 4.97i·25-s + (−1.53 + 1.53i)29-s − 3.26i·31-s + (−0.111 − 0.111i)35-s + (1.05 − 1.05i)37-s + 1.26·41-s + (−0.484 − 0.484i)43-s − 11.2·47-s + ⋯ |
L(s) = 1 | + (0.0498 + 0.0498i)5-s − 0.377·7-s + (1.08 − 1.08i)11-s + (−0.539 − 0.539i)13-s − 1.16i·17-s + (−0.695 + 0.695i)19-s + 1.37i·23-s − 0.995i·25-s + (−0.284 + 0.284i)29-s − 0.586i·31-s + (−0.0188 − 0.0188i)35-s + (0.173 − 0.173i)37-s + 0.197·41-s + (−0.0738 − 0.0738i)43-s − 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7525450587\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7525450587\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.111 - 0.111i)T + 5iT^{2} \) |
| 11 | \( 1 + (-3.61 + 3.61i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.94 + 1.94i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.79iT - 17T^{2} \) |
| 19 | \( 1 + (3.03 - 3.03i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.58iT - 23T^{2} \) |
| 29 | \( 1 + (1.53 - 1.53i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.26iT - 31T^{2} \) |
| 37 | \( 1 + (-1.05 + 1.05i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + (0.484 + 0.484i)T + 43iT^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + (4.00 + 4.00i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.61 - 7.61i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.44 - 5.44i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.897 + 0.897i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.83iT - 71T^{2} \) |
| 73 | \( 1 + 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 15.4iT - 79T^{2} \) |
| 83 | \( 1 + (7.57 + 7.57i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.139623584764357083794600624740, −7.42234045124545415618347907596, −6.58909233455091543977792858156, −5.97335407845569729677598630975, −5.25483022110273451406829635165, −4.22386713041846289194657842299, −3.44745450720215747175674002860, −2.69193517459103809322345758339, −1.44016220042765237913046450712, −0.21137640298420882936077616665,
1.47010509511709110828433288047, 2.26269719737071041683603730086, 3.40523570459297360605072529617, 4.33183226799173214205267977670, 4.76478401402040464177077884694, 5.95798931965486405041851313261, 6.71862820424911079307428519413, 7.00219617430944669750132545853, 8.108598238155853906865671279879, 8.775936351784432098202753074437