| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 3i·7-s − i·8-s − 9-s + 6.27·11-s + i·12-s + 4.27i·13-s − 3·14-s + 16-s + 5.27i·17-s − i·18-s + 4.27·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.13i·7-s − 0.353i·8-s − 0.333·9-s + 1.89·11-s + 0.288i·12-s + 1.18i·13-s − 0.801·14-s + 0.250·16-s + 1.27i·17-s − 0.235i·18-s + 0.980·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804530714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804530714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 6.27T + 11T^{2} \) |
| 13 | \( 1 - 4.27iT - 13T^{2} \) |
| 17 | \( 1 - 5.27iT - 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 29 | \( 1 + 5.54T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.274iT - 43T^{2} \) |
| 47 | \( 1 - 0.725iT - 47T^{2} \) |
| 53 | \( 1 + 4.54iT - 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.450iT - 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 17.5iT - 83T^{2} \) |
| 89 | \( 1 + 0.725T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.880140480477263852841306857185, −8.119375333223168262353271213406, −7.17132899542913607052006790607, −6.58337414686700671487589975852, −6.01340994966773131943982067030, −5.33257325303442110663106435830, −4.17836492819956887547504499313, −3.53895510065112608404191190105, −2.10770954251267891160715296125, −1.32439920199174031747617818990,
0.61089046749243415503766025482, 1.47188612187990698196618967379, 3.10515231569341738751091223940, 3.44756397554368251398436924922, 4.44230930128891865404965477109, 4.96288698051467792995368870445, 6.04911771350674451097164522529, 6.92453554937983194766125843606, 7.65270344440284417904913184173, 8.547730032956889210582858885856
