L(s) = 1 | − i·3-s + 3.46i·5-s + 7-s − 9-s − 1.46i·11-s − 2i·13-s + 3.46·15-s + 0.535·17-s − 6.92i·19-s − i·21-s + 1.46·23-s − 6.99·25-s + i·27-s + 4.92i·29-s − 10.9·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.54i·5-s + 0.377·7-s − 0.333·9-s − 0.441i·11-s − 0.554i·13-s + 0.894·15-s + 0.129·17-s − 1.58i·19-s − 0.218i·21-s + 0.305·23-s − 1.39·25-s + 0.192i·27-s + 0.915i·29-s − 1.96·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6848658263\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6848658263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 1.46iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 - 4.92iT - 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 1.07iT - 59T^{2} \) |
| 61 | \( 1 - 8.92iT - 61T^{2} \) |
| 67 | \( 1 - 2.92iT - 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−7.60695560397237731046987277293, −7.10346728365021772394382729282, −6.80651079931665702818922277803, −5.73058257035118926205451023328, −5.28570607587603427876140026760, −4.01089073467355661371277732618, −3.12636771762501517980500728571, −2.63718312255148228911989984176, −1.60398971032113328275597355286, −0.17381454260707153451727177491,
1.31266460715601620566395808577, 1.98238114420513347649981985564, 3.41741482770155045382776749723, 4.22692481987820930200355729539, 4.68787141381768481291374563778, 5.49179437509651581273510047801, 5.97435330886539878441943462981, 7.17767938062245499335509595523, 7.911392492054648129226262645091, 8.537146552438726628903713290931