L(s) = 1 | + (−0.576 + 1.29i)2-s − 3-s + (−1.33 − 1.48i)4-s + (0.576 − 1.29i)6-s + 1.97i·7-s + (2.69 − 0.867i)8-s + 9-s − 1.43i·11-s + (1.33 + 1.48i)12-s − 0.241·13-s + (−2.55 − 1.13i)14-s + (−0.430 + 3.97i)16-s + 7.38i·17-s + (−0.576 + 1.29i)18-s − 3.04i·19-s + ⋯ |
L(s) = 1 | + (−0.407 + 0.913i)2-s − 0.577·3-s + (−0.667 − 0.744i)4-s + (0.235 − 0.527i)6-s + 0.747i·7-s + (0.951 − 0.306i)8-s + 0.333·9-s − 0.431i·11-s + (0.385 + 0.429i)12-s − 0.0669·13-s + (−0.682 − 0.304i)14-s + (−0.107 + 0.994i)16-s + 1.79i·17-s + (−0.135 + 0.304i)18-s − 0.697i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0416357 + 0.547384i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0416357 + 0.547384i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.576 - 1.29i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.97iT - 7T^{2} \) |
| 11 | \( 1 + 1.43iT - 11T^{2} \) |
| 13 | \( 1 + 0.241T + 13T^{2} \) |
| 17 | \( 1 - 7.38iT - 17T^{2} \) |
| 19 | \( 1 + 3.04iT - 19T^{2} \) |
| 23 | \( 1 - 0.874iT - 23T^{2} \) |
| 29 | \( 1 - 9.07iT - 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 3.34iT - 47T^{2} \) |
| 53 | \( 1 - 9.20T + 53T^{2} \) |
| 59 | \( 1 - 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 + 4.86T + 67T^{2} \) |
| 71 | \( 1 + 8.21T + 71T^{2} \) |
| 73 | \( 1 + 4.12iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−10.76406171127108198851658645260, −10.29891275578917837694449881964, −8.949937040414259038824935878229, −8.648408908981043639484675445913, −7.42088177625559658341959099886, −6.55462898895237448984203107038, −5.70127714066814780349967657830, −5.03467278603935347831972202236, −3.66650105086487649643329325288, −1.63180324251250248226735114995,
0.38538771242626630338935628688, 1.95201572928151789800875427303, 3.43344876378660497944658171183, 4.46645372898650948216136211713, 5.38164773133632364656506531603, 6.94437067878494030250561097370, 7.54806652950994005058301993775, 8.678557615193557309162527332370, 9.783016757802585121928541165533, 10.15904752119304421374262218608