| L(s) = 1 | + 2.44i·3-s + 1.41·7-s − 2.99·9-s − 2i·11-s − 5.65i·13-s + 4.89·17-s + 6i·19-s + 3.46i·21-s + 7.07·23-s + 6.92i·29-s + 6.92·31-s + 4.89·33-s − 2.82i·37-s + 13.8·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + 1.41i·3-s + 0.534·7-s − 0.999·9-s − 0.603i·11-s − 1.56i·13-s + 1.18·17-s + 1.37i·19-s + 0.755i·21-s + 1.47·23-s + 1.28i·29-s + 1.24·31-s + 0.852·33-s − 0.464i·37-s + 2.21·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926247437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926247437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.44iT - 3T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 2.44iT - 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.751418479141308082631949145904, −8.786870489912152280692300328608, −8.179920667072441375208664513805, −7.37905276405055001039375600413, −5.95794861114106557921263760825, −5.33556892633710151402918392969, −4.69142473178184522255724791621, −3.46705258049797559335145238696, −3.09321895281421048473568916542, −1.14865351079801955579773138245,
0.957599599804535232267289757337, 1.90908187040050605445836791699, 2.83750464082889698456162455589, 4.34514828406207236746662602339, 5.10543064561347207049761334964, 6.33963196139394875969851212270, 6.89446304075254980070266331497, 7.50940059347626022226593475223, 8.280633968704939077687267340877, 9.123147105210870759975871620533
