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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.078447850\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078447850\) |
\(L(1)\) |
|
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
−10.23392878286525140248780318120, −9.729004334091381238895373971674, −8.936392079636107408664329479193, −7.84855435558716677520218045728, −7.16475291803559823994934002902, −6.27612004249513889200114167210, −4.95256425028965314608457326350, −4.02175718402967530696118647528, −3.37946706796594466844216085683, −2.14814985536626506751662612022,
1.05583541691977230752345981379, 2.39815850090055086382958484749, 3.59140465650062306201763060742, 4.75105702172542097917886065769, 5.61160677826465504215446346353, 6.86429486114435973651228308372, 7.60003251189679879365899329311, 8.181185703704613724578390242460, 9.054658721821552633669598858701, 9.714787144576502262769115616118