L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + 13-s + 0.999·15-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)25-s + 27-s − 29-s + (0.499 + 0.866i)33-s + (0.5 − 0.866i)39-s + (0.5 + 0.866i)47-s + (0.499 + 0.866i)51-s − 0.999·55-s + (0.5 + 0.866i)65-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + 13-s + 0.999·15-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)25-s + 27-s − 29-s + (0.499 + 0.866i)33-s + (0.5 − 0.866i)39-s + (0.5 + 0.866i)47-s + (0.499 + 0.866i)51-s − 0.999·55-s + (0.5 + 0.866i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.681862900\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.681862900\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.652446336320477945843565088367, −7.65222500137946550544495172951, −7.46690627729885636446035196478, −6.46720430206765895279857520882, −6.07442689358017728000071771267, −5.00004235517666970918218422334, −3.97032340608221986595279972084, −3.04221483719978103522570925985, −2.13371016779172160361029019879, −1.59228024958307104524064297409,
0.927869125450152359380604797974, 2.24601110414887581424762709628, 3.27408788186626391917586658343, 3.97712806400171567523942674410, 4.78707212293264494304231723380, 5.52467628232886686018977425399, 6.18882148716012650976353155581, 7.17316743724677732856886375801, 8.212454549756463983408855911705, 8.733614783660548093633074642694