L(s) = 1 | + 1.73i·3-s + (−0.5 − 2.59i)7-s − 2.99·9-s + 1.73i·13-s + 5.19i·19-s + (4.5 − 0.866i)21-s − 5.19i·27-s + 1.73i·31-s − 10·37-s − 2.99·39-s − 13·43-s + (−6.5 + 2.59i)49-s − 9·57-s + 15.5i·61-s + (1.49 + 7.79i)63-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (−0.188 − 0.981i)7-s − 0.999·9-s + 0.480i·13-s + 1.19i·19-s + (0.981 − 0.188i)21-s − 0.999i·27-s + 0.311i·31-s − 1.64·37-s − 0.480·39-s − 1.98·43-s + (−0.928 + 0.371i)49-s − 1.19·57-s + 1.99i·61-s + (0.188 + 0.981i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4078988828\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4078988828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 13T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5iT - 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5.19iT - 97T^{2} \) |
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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
−9.622374111942121798963163730413, −8.816880269002095787246616355723, −8.096623269098996807319859727761, −7.16754505083222596278343030046, −6.34756176471781688144033231681, −5.42216570188601923864922401393, −4.56424454436950869252315099113, −3.81104315339818470271457128910, −3.12759490888600594342797124400, −1.61479103205954372942150401786,
0.13705566650426480539799249399, 1.65673869916801412999134554978, 2.62367857746865258595303585608, 3.39358280196009456291730874820, 4.94317266087390844950543060653, 5.54906314834942799726069449773, 6.47289798513871354390315913264, 7.01023774970667470421253418041, 7.996438980255835718943383840186, 8.606399654774577926448403327291