L(s) = 1 | − 2-s + 3.40i·3-s + 4-s + (−1.28 + 1.83i)5-s − 3.40i·6-s + 2.06i·7-s − 8-s − 8.58·9-s + (1.28 − 1.83i)10-s + 3.77·11-s + 3.40i·12-s − 2.88·13-s − 2.06i·14-s + (−6.23 − 4.36i)15-s + 16-s + 5.80·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.96i·3-s + 0.5·4-s + (−0.573 + 0.819i)5-s − 1.38i·6-s + 0.779i·7-s − 0.353·8-s − 2.86·9-s + (0.405 − 0.579i)10-s + 1.13·11-s + 0.982i·12-s − 0.799·13-s − 0.551i·14-s + (−1.60 − 1.12i)15-s + 0.250·16-s + 1.40·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.111626 - 0.737662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.111626 - 0.737662i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (1.28 - 1.83i)T \) |
| 37 | \( 1 + (4.80 + 3.72i)T \) |
good | 3 | \( 1 - 3.40iT - 3T^{2} \) |
| 7 | \( 1 - 2.06iT - 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 - 0.157iT - 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 4.29iT - 29T^{2} \) |
| 31 | \( 1 + 0.425iT - 31T^{2} \) |
| 41 | \( 1 - 0.923T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.676iT - 47T^{2} \) |
| 53 | \( 1 - 9.87iT - 53T^{2} \) |
| 59 | \( 1 - 8.47iT - 59T^{2} \) |
| 61 | \( 1 + 1.23iT - 61T^{2} \) |
| 67 | \( 1 - 6.45iT - 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 + 0.980iT - 73T^{2} \) |
| 79 | \( 1 + 8.04iT - 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−11.69257623004065264740067281052, −10.68591469164344119336275381132, −10.09462537282886681796608764413, −9.218971075025095829351445611425, −8.647452727958733447195528097814, −7.36328036506570508058721442455, −6.03000084321110995197811025050, −4.97451631146306893994574908924, −3.66003666680008734498482460014, −2.87184479737424485641990223304,
0.66658622304901902266149138920, 1.61057951853498497082538792134, 3.34950857922547723215366095390, 5.25108576156108492938672530936, 6.59376640381498493478050900115, 7.24487023659216484695192395317, 7.969278488055650319238039612080, 8.697757892376873047420060770886, 9.781736900323142486128804585040, 11.28483709340208512553423515990