L(s) = 1 | + (0.5 − 1.65i)3-s − 3.31i·5-s + (−2.5 − 1.65i)9-s − 3.31i·11-s + (−5.5 − 1.65i)15-s − 3.31i·23-s − 6·25-s + (−4 + 3.31i)27-s − 5·31-s + (−5.5 − 1.65i)33-s + 7·37-s + (−5.5 + 8.29i)45-s + 6.63i·47-s + 7·49-s − 13.2i·53-s + ⋯ |
L(s) = 1 | + (0.288 − 0.957i)3-s − 1.48i·5-s + (−0.833 − 0.552i)9-s − 1.00i·11-s + (−1.42 − 0.428i)15-s − 0.691i·23-s − 1.20·25-s + (−0.769 + 0.638i)27-s − 0.898·31-s + (−0.957 − 0.288i)33-s + 1.15·37-s + (−0.819 + 1.23i)45-s + 0.967i·47-s + 49-s − 1.82i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.416826841\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.416826841\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 1.65i)T \) |
| 11 | \( 1 + 3.31iT \) |
good | 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 + 13.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 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 | \( 1 - 17T + 97T^{2} \) |
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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
−8.735542653030680697902887100002, −8.020554353859043114915957538936, −7.27844567579692207367038681363, −6.18863176910079900482594720154, −5.63725309402560693470137501505, −4.68881497114882490288591203939, −3.66593393715245845962357175117, −2.53470721568946959113476013715, −1.36535754743472841751745351922, −0.48464551870805226415211461413,
2.06134838773840352620194629119, 2.91092074279684780401366957678, 3.71491036660840051082780269267, 4.53708104967998504886270956656, 5.55411432761907979350050358359, 6.38128631614240847576100861824, 7.34503481884025642830063362000, 7.77567340058232651941051968816, 9.020535395406758595907800001202, 9.549713229132018522012750565200