| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 3i·7-s + i·8-s − 9-s − 2.77·11-s + i·12-s − 4.77i·13-s − 3·14-s + 16-s − 7.77i·17-s + i·18-s + 0.772·19-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 − 0.835·11-s + 0.288i·12-s − 1.32i·13-s − 0.801·14-s + 0.250·16-s − 1.88i·17-s + 0.235i·18-s + 0.177·19-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
| \(L(1)\) |
\(\approx\) |
\(1.057582285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057582285\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 + 4.77iT - 13T^{2} \) |
| 17 | \( 1 + 7.77iT - 17T^{2} \) |
| 19 | \( 1 - 0.772T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 + 1.77iT - 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 - 0.772iT - 43T^{2} \) |
| 47 | \( 1 + 0.227iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 7.54T + 61T^{2} \) |
| 67 | \( 1 - 7.54iT - 67T^{2} \) |
| 71 | \( 1 - 3.77T + 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 + 8.22T + 89T^{2} \) |
| 97 | \( 1 - 0.455iT - 97T^{2} \) |
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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
−7.970350446001838364142834136050, −7.48046724988700361897404422890, −6.81311697061595279826580058769, −5.62641437973552570817766773703, −5.03203241281339038326979409966, −4.10933760273608695210061244762, −3.02417666737965831720284879117, −2.54634401458542616601587730565, −1.05895620378190749750238118381, −0.36153152473413491898793648295,
1.74309218606890999731437430813, 2.77589361856562943826472469057, 3.88281240026503668013453134726, 4.58822106379877860456208792053, 5.44576166782236029087660637607, 6.03999814548599303730780442303, 6.68026471297264566823800713371, 7.79890349675756637434069724000, 8.363599190830442273418882117383, 8.993425547508902107152273707838
