L(s) = 1 | + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.22 − 0.707i)7-s + 1.41i·10-s + (0.999 − 1.73i)14-s + (0.499 + 0.866i)16-s + (0.5 + 0.866i)20-s − 1.41i·28-s + (1.22 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (1.22 + 0.707i)32-s + 1.41i·35-s − 37-s + (−0.5 − 0.866i)47-s + ⋯ |
L(s) = 1 | + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (1.22 − 0.707i)7-s + 1.41i·10-s + (0.999 − 1.73i)14-s + (0.499 + 0.866i)16-s + (0.5 + 0.866i)20-s − 1.41i·28-s + (1.22 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (1.22 + 0.707i)32-s + 1.41i·35-s − 37-s + (−0.5 − 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.542713085\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.542713085\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−8.478499749345783567591102676052, −8.073498837148660099530167640303, −7.12348144832413339363008263061, −6.53063123151449472267874977871, −5.32525680514042746651559890159, −4.84096493803381442125342042960, −3.96882884523849774906428846481, −3.41200343143198147628981986891, −2.45713765439714152096810372335, −1.46751715252266370944596190580,
1.27036646009138617738167413137, 2.58477148578749871926585032246, 3.74123514913048708029816218920, 4.53430283961575243409377519814, 5.01935891409478439701681690436, 5.58021581971990007543261638050, 6.45949723561087218192974298514, 7.31580076194902298022313467751, 8.102636883244244227897888418841, 8.557317386978878943800450016542