| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 5i·7-s − i·8-s − 9-s − i·12-s + 2i·13-s + 5·14-s + 16-s − 3i·17-s − i·18-s − 2·19-s + 5·21-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.88i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.554i·13-s + 1.33·14-s + 0.250·16-s − 0.727i·17-s − 0.235i·18-s − 0.458·19-s + 1.09·21-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
| \(L(1)\) |
\(\approx\) |
\(0.3168434061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3168434061\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.228748773206447603987545793015, −7.52303623655672188473939267393, −6.90790957916771441219695261284, −6.28992031284181088572367084469, −5.17685949026239054688768788434, −4.46046835061462749639832182058, −3.96312841482922974109616866039, −3.03584315091081225000454124690, −1.38228166563826024199385256515, −0.095772021392148762116869379072,
1.52339799204544120641488527494, 2.36710823088948219483591087991, 2.96681765471800979453144584953, 4.07143876854910272558375032741, 5.18458523872596187799710864741, 5.79471362257843304807031075609, 6.36008398282496716250461758773, 7.55530172115355737128113789591, 8.271691042226575472260952390266, 8.895639753042030860631324562129
