| L(s) = 1 | − i·3-s + 1.97·7-s − 9-s + 1.43i·11-s + 0.241i·13-s − 7.38·17-s − 3.04i·19-s − 1.97i·21-s − 0.874·23-s + i·27-s − 9.07i·29-s + 7.44·31-s + 1.43·33-s − 8.81i·37-s + 0.241·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.747·7-s − 0.333·9-s + 0.431i·11-s + 0.0669i·13-s − 1.79·17-s − 0.697i·19-s − 0.431i·21-s − 0.182·23-s + 0.192i·27-s − 1.68i·29-s + 1.33·31-s + 0.249·33-s − 1.44i·37-s + 0.0386·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366034886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366034886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 - 0.241iT - 13T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 + 3.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.874T + 23T^{2} \) |
| 29 | \( 1 + 9.07iT - 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 8.81iT - 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 3.34T + 47T^{2} \) |
| 53 | \( 1 + 9.20iT - 53T^{2} \) |
| 59 | \( 1 - 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 - 4.86iT - 67T^{2} \) |
| 71 | \( 1 - 8.21T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.626871945371609627608359994449, −7.981640137222845524427542128292, −7.12248893949799365691824878013, −6.56422007534609946323865576388, −5.62273943384332173881528038036, −4.66804506017485769732315634317, −4.03555062667602548016361077154, −2.50726043797336458250922734755, −1.96168286781778505622671664643, −0.46141658196001541960918339035,
1.35396195296306180458107768084, 2.57150300446558667187903864954, 3.55643258260277998975592899581, 4.60255221067195535888485203110, 4.98941786724535371820452670611, 6.16111742407887903400389093382, 6.73904712435935457343399167860, 7.959774512598620488365756522282, 8.402379364201089068469835048619, 9.163114590038288257398628073858
