| L(s) = 1 | + 1.73i·3-s − 2.23·5-s − 2.64i·7-s − 2.99·9-s − 5.10i·11-s − 12.4·13-s − 3.87i·15-s − 26.0·17-s + 15.5i·19-s + 4.58·21-s + 18.3i·23-s + 5.00·25-s − 5.19i·27-s + 46.5·29-s − 2.63i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447·5-s − 0.377i·7-s − 0.333·9-s − 0.464i·11-s − 0.955·13-s − 0.258i·15-s − 1.53·17-s + 0.820i·19-s + 0.218·21-s + 0.797i·23-s + 0.200·25-s − 0.192i·27-s + 1.60·29-s − 0.0850i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.334047711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.334047711\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 + 5.10iT - 121T^{2} \) |
| 13 | \( 1 + 12.4T + 169T^{2} \) |
| 17 | \( 1 + 26.0T + 289T^{2} \) |
| 19 | \( 1 - 15.5iT - 361T^{2} \) |
| 23 | \( 1 - 18.3iT - 529T^{2} \) |
| 29 | \( 1 - 46.5T + 841T^{2} \) |
| 31 | \( 1 + 2.63iT - 961T^{2} \) |
| 37 | \( 1 - 28.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 61.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 67.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 0.353iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 23.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4.09T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 99.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 96.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 97.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 94.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 3.09T + 9.40e3T^{2} \) |
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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
−9.228946488469108369008865222596, −8.387880501198737421178296639338, −7.66472283287716301804147803028, −6.78865863603900202078809114069, −5.90978049394854395548266841670, −4.84049478187130403875251135476, −4.22555266072361099789381947353, −3.29146415450271063379772599444, −2.21029484994047683872091877862, −0.54856700148899188097454186074,
0.70098836997958885540844735988, 2.25879533753591065323670560642, 2.82621933933569663533148686143, 4.43033382098300841093444876823, 4.81806716678602876463145223245, 6.18556786344140822312008357561, 6.77744422928284081485379145139, 7.54453911353736736743158831546, 8.347054933266373676411339872551, 9.061252661402600342814436606344
