L(s) = 1 | + (−1.52 + 1.63i)5-s − 1.80i·7-s + 2.90·11-s − 2.25i·13-s + 2.14i·17-s − 0.339·19-s + i·23-s + (−0.369 − 4.98i)25-s − 5.60·29-s + 5.92·31-s + (2.95 + 2.74i)35-s + 8.98i·37-s − 1.89·41-s − 9.47i·43-s − 7.83i·47-s + ⋯ |
L(s) = 1 | + (−0.680 + 0.732i)5-s − 0.682i·7-s + 0.876·11-s − 0.624i·13-s + 0.520i·17-s − 0.0779·19-s + 0.208i·23-s + (−0.0738 − 0.997i)25-s − 1.04·29-s + 1.06·31-s + (0.499 + 0.464i)35-s + 1.47i·37-s − 0.295·41-s − 1.44i·43-s − 1.14i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.473944840\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.473944840\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.52 - 1.63i)T \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 1.80iT - 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 + 2.25iT - 13T^{2} \) |
| 17 | \( 1 - 2.14iT - 17T^{2} \) |
| 19 | \( 1 + 0.339T + 19T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 - 8.98iT - 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 + 9.47iT - 43T^{2} \) |
| 47 | \( 1 + 7.83iT - 47T^{2} \) |
| 53 | \( 1 + 6.47iT - 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 9.25iT - 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.94T + 79T^{2} \) |
| 83 | \( 1 + 5.37iT - 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2.43iT - 97T^{2} \) |
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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
−8.247909338049063202914973104408, −7.55452555275239321089400885276, −6.89009097379058855244734317520, −6.33859229013356331204506925670, −5.37592537049983592706060068387, −4.32463876784982910279422770783, −3.72751562856158620341888133517, −3.05031680884573032933871447537, −1.80628131994884161739779975848, −0.53310790546795136910185305106,
0.928430532343039732031306794517, 2.01051046746176026339970252574, 3.10902490547772127034769617535, 4.10502513252580208626520118982, 4.60333350672764029082833808229, 5.54373895676632044676462886900, 6.26060948952315614431366070530, 7.12032179424065685442520790761, 7.83027610713300746089627486430, 8.566532564546566779201218168567