| L(s) = 1 | − 0.732·3-s − 2.73i·5-s + (−2 + 1.73i)7-s − 2.46·9-s + 1.46i·11-s + 1.26i·13-s + 2i·15-s − 4i·17-s + 4.73·19-s + (1.46 − 1.26i)21-s + 1.46i·23-s − 2.46·25-s + 4·27-s − 6.92·29-s − 6.92·31-s + ⋯ |
| L(s) = 1 | − 0.422·3-s − 1.22i·5-s + (−0.755 + 0.654i)7-s − 0.821·9-s + 0.441i·11-s + 0.351i·13-s + 0.516i·15-s − 0.970i·17-s + 1.08·19-s + (0.319 − 0.276i)21-s + 0.305i·23-s − 0.492·25-s + 0.769·27-s − 1.28·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9298947460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9298947460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 1.26iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 9.46iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 5.66iT - 61T^{2} \) |
| 67 | \( 1 - 9.46iT - 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 - 1.07iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.329294912132447102928164515233, −8.827713195988259462508235181312, −7.82821601224652404569232848593, −6.97941627640599550211827587177, −5.88766396042881157726493356758, −5.39836239093837971827370452998, −4.66189211065899043038366722962, −3.44806444111806495987702984262, −2.38912370413532841440509019205, −0.933063568419090143709175370031,
0.46642505627728100919598657793, 2.32173005205651868303317773901, 3.38171630551222136045051876527, 3.83164743235685588504676658622, 5.54838507311446131940763750617, 5.83961248398553963860705521068, 6.94749501947458280366255456090, 7.28991879908038477610160289985, 8.391127539797871198993387374514, 9.267445422993078116137357611485
