L(s) = 1 | + (0.866 + 0.5i)2-s + (0.991 − 0.130i)3-s + (0.499 + 0.866i)4-s + (0.923 + 0.382i)6-s + 0.999i·8-s + (0.965 − 0.258i)9-s + (−1.22 + 0.707i)11-s + (0.608 + 0.793i)12-s + (−0.5 + 0.866i)16-s + (−0.923 − 1.60i)17-s + (0.965 + 0.258i)18-s + (−0.662 − 0.382i)19-s − 1.41·22-s + (0.130 + 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.991 − 0.130i)3-s + (0.499 + 0.866i)4-s + (0.923 + 0.382i)6-s + 0.999i·8-s + (0.965 − 0.258i)9-s + (−1.22 + 0.707i)11-s + (0.608 + 0.793i)12-s + (−0.5 + 0.866i)16-s + (−0.923 − 1.60i)17-s + (0.965 + 0.258i)18-s + (−0.662 − 0.382i)19-s − 1.41·22-s + (0.130 + 0.991i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.177450315\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.177450315\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.84T + T^{2} \) |
| 89 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 0.765iT - T^{2} \) |
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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
−9.998409615839924992816174008060, −9.033903324005790650506444041439, −8.301775646923199554949197382445, −7.38012488776992080600317370556, −7.04731016343117792291615220093, −5.84908045460045185878652982551, −4.71241293402517279402986139957, −4.19420749039456483971863135388, −2.72852581456127403536610247468, −2.36531938906347591253055447549,
1.79123172966811295921957262070, 2.68036814911113335927034453918, 3.68450258381241490873032178124, 4.38513234254987449020773655164, 5.52439772870617366434208850944, 6.33197752842882123838850477479, 7.45979121386736352584169872112, 8.268990322634649869411046069351, 9.069536942660358507897657330812, 10.07940954454811518370876304604