| L(s) = 1 | − 1.82i·2-s − i·3-s − 1.33·4-s − 1.82·6-s + 1.44i·7-s − 1.20i·8-s − 9-s − 2.12·11-s + 1.33i·12-s + 5.70i·13-s + 2.64·14-s − 4.88·16-s + 4.15i·17-s + 1.82i·18-s − 1.70·19-s + ⋯ |
| L(s) = 1 | − 1.29i·2-s − 0.577i·3-s − 0.669·4-s − 0.745·6-s + 0.546i·7-s − 0.427i·8-s − 0.333·9-s − 0.641·11-s + 0.386i·12-s + 1.58i·13-s + 0.705·14-s − 1.22·16-s + 1.00i·17-s + 0.430i·18-s − 0.390·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9689806576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9689806576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.82iT - 2T^{2} \) |
| 7 | \( 1 - 1.44iT - 7T^{2} \) |
| 11 | \( 1 + 2.12T + 11T^{2} \) |
| 13 | \( 1 - 5.70iT - 13T^{2} \) |
| 17 | \( 1 - 4.15iT - 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 - 0.323iT - 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 8.45T + 31T^{2} \) |
| 37 | \( 1 - 1.75iT - 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 12.5iT - 47T^{2} \) |
| 53 | \( 1 + 8.34iT - 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 + 7.13iT - 67T^{2} \) |
| 71 | \( 1 - 2.67T + 71T^{2} \) |
| 73 | \( 1 + 6.28iT - 73T^{2} \) |
| 79 | \( 1 + 8.37T + 79T^{2} \) |
| 83 | \( 1 - 14.5iT - 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 8.55iT - 97T^{2} \) |
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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
−9.311583463731118562402977254621, −8.669144433037871422233821976224, −7.79412720848500406354970030109, −6.70674878328043610067720623170, −6.21785293720125615665495932346, −4.93429490256357414693855870591, −4.05553575788379281405249118429, −3.01045604525792667646644943531, −2.14051143856315248636067295966, −1.41877274397746503838811789098,
0.34325584816555734179749571956, 2.45931674743681622299338024079, 3.47976790027310219839269514256, 4.67515796442580037598078624985, 5.33293798444384655072142979551, 5.90705274919238331383490853684, 7.11081381363683053149641769798, 7.41872197510806929573549277246, 8.446502228422436971813034908765, 8.815190905273289689030756545479
