L(s) = 1 | + 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·5-s − 1.41i·6-s + 7-s − 2.00·10-s + 1.00·12-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 17-s + 19-s − 1.41i·20-s − 21-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 3-s − 1.00·4-s + 1.41i·5-s − 1.41i·6-s + 7-s − 2.00·10-s + 1.00·12-s + 1.41i·14-s − 1.41i·15-s − 0.999·16-s − 17-s + 19-s − 1.41i·20-s − 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6523332469\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6523332469\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 563 | \( 1 + T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - 1.41iT - T^{2} \) |
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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
−11.28187127132090665349358956268, −10.81729902141978609999300877975, −9.620196492387458257340825716854, −8.331956355553905437178219978167, −7.58954304067215796392683223779, −6.78642278381194156730978081496, −6.07782034797219569215891263814, −5.36190289739853469999498574010, −4.28127219713578690712690574835, −2.47192466332174092058953421171,
0.925630044959445458646108862088, 2.09780978497663742922633009838, 3.86133284679924835613565817504, 4.89426354768251895145989938166, 5.37499563720585411900221105330, 6.81321085057391180508905555377, 8.283771820274635487078762950557, 8.923553383035821704262623852138, 9.935318588558455993266471417697, 10.83395599285513975218397484582