L(s) = 1 | − i·3-s + (1.45 − 1.69i)5-s − 0.243i·7-s − 9-s + 0.348·11-s + 2.62i·13-s + (−1.69 − 1.45i)15-s + 1.29i·17-s + 1.36·19-s − 0.243·21-s − 0.600i·23-s + (−0.768 − 4.94i)25-s + i·27-s − 3.99·29-s − 3.23·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.650 − 0.759i)5-s − 0.0919i·7-s − 0.333·9-s + 0.105·11-s + 0.728i·13-s + (−0.438 − 0.375i)15-s + 0.313i·17-s + 0.313·19-s − 0.0530·21-s − 0.125i·23-s + (−0.153 − 0.988i)25-s + 0.192i·27-s − 0.742·29-s − 0.581·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.674007456\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.674007456\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.45 + 1.69i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 + 0.243iT - 7T^{2} \) |
| 11 | \( 1 - 0.348T + 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 - 1.29iT - 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 0.600iT - 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 7.96iT - 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 + 8.24iT - 43T^{2} \) |
| 47 | \( 1 + 9.05iT - 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 6.00T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 + 8.87iT - 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 - 0.828iT - 97T^{2} \) |
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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
−8.305795872361121820527203268687, −7.30916602486203131698250849964, −6.83588300811870067518277354373, −5.79772923783197995289712339993, −5.44491991927545008833871406525, −4.38545633426045524540569733930, −3.60246876792657419066380403355, −2.25280534915866813532777022035, −1.68970625266392880859543982205, −0.47455804217907918766702985701,
1.31581286001372227677216047770, 2.60466474445421424270572958765, 3.14148779880723777450476326238, 4.11013977887788071844304438004, 5.06801731818941028265367042498, 5.77177593930676452609797995188, 6.34395510939961900539388920863, 7.32858802830909041655525329031, 7.86164980833785124065103902545, 8.946360779214847982440811266421