| L(s) = 1 | + i·3-s + 2.82i·5-s + 2.82·7-s − 9-s − 4i·11-s + 5.65i·13-s − 2.82·15-s − 2·17-s + 4i·19-s + 2.82i·21-s − 5.65·23-s − 3.00·25-s − i·27-s + 2.82i·29-s + 8.48·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.26i·5-s + 1.06·7-s − 0.333·9-s − 1.20i·11-s + 1.56i·13-s − 0.730·15-s − 0.485·17-s + 0.917i·19-s + 0.617i·21-s − 1.17·23-s − 0.600·25-s − 0.192i·27-s + 0.525i·29-s + 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.998011 + 0.998011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.998011 + 0.998011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−11.33361748402886339195337101211, −10.81651310302083702446349800780, −9.914307836848614405954978480707, −8.763188948667387345045984116745, −7.955343462929157905281491771570, −6.73331959112034251565986066026, −5.89445475670083262665834723464, −4.51608294301034087429074745230, −3.52284000970033550234053273583, −2.11078420427884692762323138139,
1.02510739554723904828571736839, 2.40438796828212904038147726604, 4.46856019048839960304531083966, 5.05460623651132301736697090399, 6.26814282724700982845983465206, 7.78189179184266387615888390757, 8.049838974768667062494455871664, 9.151892903387540535683859369935, 10.15945751752905598937381700221, 11.30573323030386268123070068411
