L(s) = 1 | + (−0.575 + 1.29i)2-s + (0.978 + 0.207i)3-s + (−0.669 − 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.831 + 1.14i)6-s + (0.294 + 1.38i)7-s + (0.913 + 0.406i)9-s − 1.41i·10-s + (−0.500 − 0.866i)12-s + (−1.95 − 0.415i)14-s + (−0.978 + 0.207i)15-s + (0.104 − 0.994i)16-s + (−1.05 + 0.946i)18-s + (0.913 + 0.406i)20-s + 1.41i·21-s + ⋯ |
L(s) = 1 | + (−0.575 + 1.29i)2-s + (0.978 + 0.207i)3-s + (−0.669 − 0.743i)4-s + (−0.913 + 0.406i)5-s + (−0.831 + 1.14i)6-s + (0.294 + 1.38i)7-s + (0.913 + 0.406i)9-s − 1.41i·10-s + (−0.500 − 0.866i)12-s + (−1.95 − 0.415i)14-s + (−0.978 + 0.207i)15-s + (0.104 − 0.994i)16-s + (−1.05 + 0.946i)18-s + (0.913 + 0.406i)20-s + 1.41i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8793508330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8793508330\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.575 - 1.29i)T + (-0.669 - 0.743i)T^{2} \) |
| 5 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 7 | \( 1 + (-0.294 - 1.38i)T + (-0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.294 + 1.38i)T + (-0.913 + 0.406i)T^{2} \) |
| 31 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (-1.40 - 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (-1.40 - 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \) |
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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
−9.965292405923068204404391611433, −9.304488625072442060966143567117, −8.564171079306512714640942704684, −7.992658747791418084565942153793, −7.48827618228487643168674014193, −6.48246619293802789205989219387, −5.58424082964123707553266700035, −4.49902058371925423195636939778, −3.29032131056179388080665998022, −2.28767458697672807493993280846,
0.909089372346202365226210124633, 1.99667463695322544675256769664, 3.44692344098981953196377601846, 3.78664664712424705844895184818, 4.84443291699099181559836304970, 6.74517201400762820774451721109, 7.47664393618516720776555204811, 8.296123971406569424829593808890, 8.847425821050452602208541596210, 9.793135562332075341247413809168