L(s) = 1 | − 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 1.41i·5-s + 2.00·6-s − 1.00·9-s − 2.00·10-s − 11-s − 1.41i·12-s + 2.00·15-s − 0.999·16-s − 17-s + 1.41i·18-s − 1.41i·19-s + 1.41i·20-s + ⋯ |
L(s) = 1 | − 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 1.41i·5-s + 2.00·6-s − 1.00·9-s − 2.00·10-s − 11-s − 1.41i·12-s + 2.00·15-s − 0.999·16-s − 17-s + 1.41i·18-s − 1.41i·19-s + 1.41i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7085197646\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7085197646\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 233 | \( 1 - T \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−8.966308471757042360398399669638, −8.531493529962721411274021462105, −7.39169254687569231610171592698, −5.97810237262983055390819926905, −5.02538466275586459080282852221, −4.47807619637250736327595134535, −4.02889128610351794862898201543, −2.88022517795644054179448284056, −2.01167708205642339916872271798, −0.42656929302875800099008565060,
1.94669311806842822348200736691, 2.64741712167278805447573544247, 3.91096781329385457098304243135, 5.28800620688109138976054973666, 6.03133522837937432936503459265, 6.50892096593132392100438424575, 7.24586468610773280239023326767, 7.56992681696179587969796202837, 8.198120055933097180651501151626, 9.024199886102955549724432401123