L(s) = 1 | + (−0.5 + 0.866i)3-s + 11-s + (−1 + 1.73i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)41-s + (1 − 1.73i)43-s + 49-s + (−0.999 − 1.73i)51-s − 0.999·57-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)67-s + (0.5 − 0.866i)73-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + 11-s + (−1 + 1.73i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)41-s + (1 − 1.73i)43-s + 49-s + (−0.999 − 1.73i)51-s − 0.999·57-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)67-s + (0.5 − 0.866i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8042628884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8042628884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−10.72838366275460956136030641639, −10.41801524758241042228149675412, −9.369956775774070160574742727137, −8.592155664383104785485620073556, −7.50191829759378837732288657574, −6.30017884713119662982360887260, −5.61262967968093306371810311304, −4.29070765762225192662841796841, −3.83026987453406008490100688988, −1.92319419153140999332962519896,
1.12585683036847704027328366797, 2.67437094137505263442893173742, 4.14714638658681278038487665856, 5.27092499718470067659696579444, 6.41371965965863800961026966213, 6.98546810395350658811309259632, 7.77289761096655262451228694979, 9.265106896335518555685611048344, 9.441622027031549088590967758695, 11.10447021548712800271749983443