L(s) = 1 | + (−1.15 − 0.814i)2-s + (−0.887 − 1.48i)3-s + (0.672 + 1.88i)4-s + (−0.185 + 2.44i)6-s + 0.797·7-s + (0.757 − 2.72i)8-s + (−1.42 + 2.64i)9-s − 0.320i·11-s + (2.20 − 2.67i)12-s + 4.30·13-s + (−0.921 − 0.649i)14-s + (−3.09 + 2.53i)16-s − 2.57·17-s + (3.79 − 1.89i)18-s + 6.10·19-s + ⋯ |
L(s) = 1 | + (−0.817 − 0.576i)2-s + (−0.512 − 0.858i)3-s + (0.336 + 0.941i)4-s + (−0.0756 + 0.997i)6-s + 0.301·7-s + (0.267 − 0.963i)8-s + (−0.474 + 0.880i)9-s − 0.0966i·11-s + (0.636 − 0.771i)12-s + 1.19·13-s + (−0.246 − 0.173i)14-s + (−0.773 + 0.633i)16-s − 0.624·17-s + (0.894 − 0.446i)18-s + 1.40·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.543083 - 0.661825i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543083 - 0.661825i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.15 + 0.814i)T \) |
| 3 | \( 1 + (0.887 + 1.48i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.797T + 7T^{2} \) |
| 11 | \( 1 + 0.320iT - 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 + 3.13iT - 23T^{2} \) |
| 29 | \( 1 - 8.79T + 29T^{2} \) |
| 31 | \( 1 + 9.90iT - 31T^{2} \) |
| 37 | \( 1 + 8.49T + 37T^{2} \) |
| 41 | \( 1 - 5.28iT - 41T^{2} \) |
| 43 | \( 1 - 2.97iT - 43T^{2} \) |
| 47 | \( 1 + 6.56iT - 47T^{2} \) |
| 53 | \( 1 + 3.94iT - 53T^{2} \) |
| 59 | \( 1 + 12.4iT - 59T^{2} \) |
| 61 | \( 1 + 8.83iT - 61T^{2} \) |
| 67 | \( 1 - 4.66iT - 67T^{2} \) |
| 71 | \( 1 - 3.43T + 71T^{2} \) |
| 73 | \( 1 - 1.43iT - 73T^{2} \) |
| 79 | \( 1 + 2.89iT - 79T^{2} \) |
| 83 | \( 1 - 3.37T + 83T^{2} \) |
| 89 | \( 1 - 13.7iT - 89T^{2} \) |
| 97 | \( 1 + 4.26iT - 97T^{2} \) |
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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
−10.59541905758076265719666768941, −9.611536598260978520099817180313, −8.458404159811026677062091481909, −8.022821437694501384250072188708, −6.91035679606041430921773679553, −6.19504926290954490257389823601, −4.82157864845146503023482396936, −3.34805348885681852430680306509, −2.03569870843319648088644928773, −0.810269246833016453136966247583,
1.23300653408837132333477135808, 3.25013668174206428466726303444, 4.68325952447146291257225555724, 5.50336603366687417276603643274, 6.40971177816757588228333371379, 7.33075509524371926070328065245, 8.611585418445861176777328999109, 8.996243461111399483162872262098, 10.11633608651929867287901467827, 10.64888457611990846481498640786