L(s) = 1 | + i·3-s + (0.5 − 0.866i)5-s + i·7-s − 9-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s − 21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)33-s + (0.866 + 0.5i)35-s + ⋯ |
L(s) = 1 | + i·3-s + (0.5 − 0.866i)5-s + i·7-s − 9-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s − 21-s + (0.866 + 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)33-s + (0.866 + 0.5i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.034902884\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.034902884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.04403129964175742854381383704, −9.566106224549920514142174500437, −8.718419206193694609809368222391, −8.345310026683079353118656003915, −6.84309754018777651146141566271, −5.71016915359737608816061050652, −5.16168102156128327439947338625, −4.38466232835993860412268890814, −3.08867158805419358669586398597, −1.90754303632115724833325167887,
1.05403313853153721944121648055, 2.68594897231513677829888752594, 3.25635360403316347372738497227, 4.94526749195760034311737632934, 5.91133315122096204186401369294, 6.71479167669262935534432323419, 7.41448632525629595551325358937, 8.071638510828703625140363594402, 9.083582176667505874723310378852, 10.30708534121523655829777156087