L(s) = 1 | − 0.554·3-s + 4.57i·5-s + 2.64i·7-s − 8.69·9-s − 15.7·11-s + 8.57i·13-s − 2.53i·15-s + 28.3·17-s + 6.33·19-s − 1.46i·21-s + 31.0i·23-s + 4.05·25-s + 9.81·27-s − 0.846i·29-s + 21.6i·31-s + ⋯ |
L(s) = 1 | − 0.184·3-s + 0.915i·5-s + 0.377i·7-s − 0.965·9-s − 1.43·11-s + 0.659i·13-s − 0.169i·15-s + 1.66·17-s + 0.333·19-s − 0.0698i·21-s + 1.35i·23-s + 0.162·25-s + 0.363·27-s − 0.0291i·29-s + 0.697i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2830899696\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2830899696\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 3 | \( 1 + 0.554T + 9T^{2} \) |
| 5 | \( 1 - 4.57iT - 25T^{2} \) |
| 11 | \( 1 + 15.7T + 121T^{2} \) |
| 13 | \( 1 - 8.57iT - 169T^{2} \) |
| 17 | \( 1 - 28.3T + 289T^{2} \) |
| 19 | \( 1 - 6.33T + 361T^{2} \) |
| 23 | \( 1 - 31.0iT - 529T^{2} \) |
| 29 | \( 1 + 0.846iT - 841T^{2} \) |
| 31 | \( 1 - 21.6iT - 961T^{2} \) |
| 37 | \( 1 - 33.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 66.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 44.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 5.80T + 3.48e3T^{2} \) |
| 61 | \( 1 + 52.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 117.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 81.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 57.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 89.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44T + 9.40e3T^{2} \) |
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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
−9.772177795743176088538410477091, −8.700161679271210726149554688903, −7.965486133780127228073740811324, −7.26327705440997422269593898598, −6.36403278393691310286355870112, −5.47282305016028842563634155897, −5.02318460513406837584886160891, −3.23583500997218240039798817529, −3.07620286292336150794311298990, −1.70863459729737546247399340908,
0.085346640442176882852302135767, 1.02078915903581091519949678320, 2.58263382501433454249003172484, 3.39730028786017277342229358647, 4.71748606665839718645234246237, 5.35302160834667563224909038418, 5.88648589129951081459675987336, 7.14490039019688954740033184684, 8.155081197557696410101506960361, 8.269716589839348014620314999159