| L(s) = 1 | + 0.585i·3-s + 9.03·5-s − 2.64i·7-s + 8.65·9-s − 12.4i·11-s + 9.03·13-s + 5.29i·15-s + 12.3·17-s + 28.8i·19-s + 1.54·21-s + 24.6i·23-s + 56.5·25-s + 10.3i·27-s − 22.4·29-s − 16.7i·31-s + ⋯ |
| L(s) = 1 | + 0.195i·3-s + 1.80·5-s − 0.377i·7-s + 0.961·9-s − 1.13i·11-s + 0.694·13-s + 0.352i·15-s + 0.726·17-s + 1.51i·19-s + 0.0738·21-s + 1.07i·23-s + 2.26·25-s + 0.383i·27-s − 0.774·29-s − 0.541i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.533224229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.533224229\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 - 0.585iT - 9T^{2} \) |
| 5 | \( 1 - 9.03T + 25T^{2} \) |
| 11 | \( 1 + 12.4iT - 121T^{2} \) |
| 13 | \( 1 - 9.03T + 169T^{2} \) |
| 17 | \( 1 - 12.3T + 289T^{2} \) |
| 19 | \( 1 - 28.8iT - 361T^{2} \) |
| 23 | \( 1 - 24.6iT - 529T^{2} \) |
| 29 | \( 1 + 22.4T + 841T^{2} \) |
| 31 | \( 1 + 16.7iT - 961T^{2} \) |
| 37 | \( 1 - 16.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 6.97T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 6.19iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.01T + 2.80e3T^{2} \) |
| 59 | \( 1 + 30.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 15.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 78.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 17.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 46.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 81.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 40.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 164.T + 9.40e3T^{2} \) |
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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
−9.439255134613437965520457443410, −8.361660164994555847117232100684, −7.53824022386722672358891356013, −6.45858383538475013315777071613, −5.84069628561782403170435334201, −5.31791086029819426988563847109, −3.99211907596752207876543966826, −3.19711129858355616798178671284, −1.78311234396837094631928787505, −1.16936776573594438803545203241,
1.15053616377806394726625332096, 1.99782309880377823547633040802, 2.77976745193571899978394696516, 4.30331773816389276354204611390, 5.12383397320545456401329533299, 5.89057148454577788314617776437, 6.75606570373900684772154962474, 7.21211819579274202071526500913, 8.523992989986315052392941516252, 9.236784481244748525447455985185
