L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + 5-s − 1.00·6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)10-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s − 13-s + 1.00i·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + (1 − i)17-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + 5-s − 1.00·6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)10-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)12-s − 13-s + 1.00i·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + (1 − i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.022755156\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.022755156\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| 29 | \( 1 - iT \) |
good | 3 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08319997694114272629992948091, −9.721794958340751009354974684343, −9.125096894297901045544196515858, −7.904910417949662641015054546629, −7.47142156904223783992060916008, −6.36039612228367439914921242211, −5.11498285873664415564594089052, −4.68123959797184920483666777913, −2.95857849200810114779821280370, −1.66171220104048561339524486419,
1.75893815152060158962147917292, 2.27157755513400655965312707505, 3.33641095746897987665930129323, 5.03127156388103948506547684377, 5.93324406799304452702936675084, 7.34876010342720016161805605762, 7.950695874935556886706500063691, 8.637434609002580492392869229601, 9.392522819654094477135360748386, 10.33433119868509475516978713060