L(s) = 1 | + (−0.987 + 0.156i)2-s + (0.156 − 0.987i)3-s + (0.951 − 0.309i)4-s + i·6-s + (−0.891 + 0.453i)8-s + (−0.951 − 0.309i)9-s + (−1.29 + 0.101i)11-s + (−0.156 − 0.987i)12-s + (0.809 − 0.587i)16-s + (1.20 − 1.40i)17-s + (0.987 + 0.156i)18-s + (−0.453 − 1.89i)19-s + (1.26 − 0.303i)22-s + (0.309 + 0.951i)24-s + (−0.587 − 0.809i)25-s + ⋯ |
L(s) = 1 | + (−0.987 + 0.156i)2-s + (0.156 − 0.987i)3-s + (0.951 − 0.309i)4-s + i·6-s + (−0.891 + 0.453i)8-s + (−0.951 − 0.309i)9-s + (−1.29 + 0.101i)11-s + (−0.156 − 0.987i)12-s + (0.809 − 0.587i)16-s + (1.20 − 1.40i)17-s + (0.987 + 0.156i)18-s + (−0.453 − 1.89i)19-s + (1.26 − 0.303i)22-s + (0.309 + 0.951i)24-s + (−0.587 − 0.809i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5515709021\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5515709021\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 3 | \( 1 + (-0.156 + 0.987i)T \) |
| 41 | \( 1 + (-0.453 + 0.891i)T \) |
good | 5 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + (-0.453 - 0.891i)T^{2} \) |
| 11 | \( 1 + (1.29 - 0.101i)T + (0.987 - 0.156i)T^{2} \) |
| 13 | \( 1 + (0.891 + 0.453i)T^{2} \) |
| 17 | \( 1 + (-1.20 + 1.40i)T + (-0.156 - 0.987i)T^{2} \) |
| 19 | \( 1 + (0.453 + 1.89i)T + (-0.891 + 0.453i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.156 - 0.987i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (-0.183 - 1.16i)T + (-0.951 + 0.309i)T^{2} \) |
| 47 | \( 1 + (0.453 - 0.891i)T^{2} \) |
| 53 | \( 1 + (-0.156 + 0.987i)T^{2} \) |
| 59 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 67 | \( 1 + (1.98 + 0.156i)T + (0.987 + 0.156i)T^{2} \) |
| 71 | \( 1 + (0.987 - 0.156i)T^{2} \) |
| 73 | \( 1 + (-1.14 - 1.14i)T + iT^{2} \) |
| 79 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - 0.618iT - T^{2} \) |
| 89 | \( 1 + (-0.0819 + 0.133i)T + (-0.453 - 0.891i)T^{2} \) |
| 97 | \( 1 + (0.0366 - 0.465i)T + (-0.987 - 0.156i)T^{2} \) |
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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
−9.758128742567770868334299899155, −9.035760454622532234946022340876, −8.115808323047208071032618313927, −7.52418678000988443355885276191, −6.89861871060567017705903882077, −5.88382683180957757774622016501, −5.00378348533960434786210153324, −2.93155919767655518622064597354, −2.35073133612271114269729541947, −0.68031397855853713896612268016,
1.88738505720139685521230975679, 3.18505980882918029433085515407, 3.95391695189209167574991302925, 5.53439912318798759467292032028, 5.99346682112592329854553330386, 7.59902156198885588727602503301, 8.086087239507333700600748570479, 8.791067181542737566296968711843, 9.884298554219761759885913538800, 10.31073283010945651172287585952