| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.388072296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.388072296\) |
| \(L(\frac{3}{2})\) |
|
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
−8.695336139966972663594498173919, −8.108356686169501421398428862064, −7.54439147860787464234047880353, −6.70079222639323946846969264362, −5.87395015948857977393816203106, −5.17673283889535506555416819831, −3.79557429084374096162018441173, −2.67627034417922973715446433289, −1.67115220731945151723606599809, −0.53112305967938897472796508173,
1.76453684665982729058721692108, 3.00343674054883733757813618222, 4.17237461924182306661152297575, 4.69473285539727722197115074405, 5.25458779232025572559490467718, 6.76968440910288012950796567055, 7.12699377897739351176450563928, 8.643985497340552105026267577366, 9.099129091832756119870842652159, 9.552995538655591881121420653876
