| L(s) = 1 | − 3i·2-s + 3i·3-s − 4-s + 9·6-s − 7i·7-s − 21i·8-s − 9·9-s + 11·11-s − 3i·12-s + 16i·13-s − 21·14-s − 71·16-s + 21i·17-s + 27i·18-s − 125·19-s + ⋯ |
| L(s) = 1 | − 1.06i·2-s + 0.577i·3-s − 0.125·4-s + 0.612·6-s − 0.377i·7-s − 0.928i·8-s − 0.333·9-s + 0.301·11-s − 0.0721i·12-s + 0.341i·13-s − 0.400·14-s − 1.10·16-s + 0.299i·17-s + 0.353i·18-s − 1.50·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3iT - 8T^{2} \) |
| 7 | \( 1 + 7iT - 343T^{2} \) |
| 13 | \( 1 - 16iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 21iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 125T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 186T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58T + 2.97e4T^{2} \) |
| 37 | \( 1 + 253iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 63T + 6.89e4T^{2} \) |
| 43 | \( 1 - 100iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 219iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 192iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 249T + 2.05e5T^{2} \) |
| 61 | \( 1 + 64T + 2.26e5T^{2} \) |
| 67 | \( 1 - 272iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 645T + 3.57e5T^{2} \) |
| 73 | \( 1 - 112iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 509T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 756T + 7.04e5T^{2} \) |
| 97 | \( 1 - 839iT - 9.12e5T^{2} \) |
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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
−9.457934176600022399068906861059, −8.812013496463059194284410096585, −7.55016914545162682355472515475, −6.63855070864133296465718794923, −5.61931890097627735062426332253, −4.18698906397365303528222045677, −3.80948539161061902048809567135, −2.52696007623732189304681306258, −1.50237790942538438773731899179, 0,
1.78654587583691056466323434128, 2.80895376000091989098846280159, 4.37009259755175689019163108613, 5.48662081952201599958294920305, 6.21771133437610744721269914907, 6.89707426528867103012573556693, 7.74114880501849830680047366764, 8.491962787291807901534593453650, 9.155790297706069046263117583171
