L(s) = 1 | − 3-s − 5-s + 9-s − 11-s + 4·13-s + 15-s − 3·17-s + 7·19-s − 9·23-s + 25-s − 27-s − 3·29-s − 2·31-s + 33-s − 4·37-s − 4·39-s + 6·41-s − 43-s − 45-s + 6·47-s + 3·51-s + 3·53-s + 55-s − 7·57-s + 9·59-s + 61-s − 4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.727·17-s + 1.60·19-s − 1.87·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.152·43-s − 0.149·45-s + 0.875·47-s + 0.420·51-s + 0.412·53-s + 0.134·55-s − 0.927·57-s + 1.17·59-s + 0.128·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.52581548975454, −14.84235144362981, −14.16766739367647, −13.64556874083923, −13.29662290243541, −12.55512061520299, −12.08357931802826, −11.48178544728753, −11.25994877268757, −10.49280660970112, −10.12306279808395, −9.389991156597378, −8.825044944647968, −8.238682347349025, −7.547742716759507, −7.231082823629955, −6.385783016248032, −5.846578122655711, −5.418493830140939, −4.629527666264413, −3.910575467696806, −3.566742989544581, −2.583172540894343, −1.752765374845802, −0.9192808629369233, 0,
0.9192808629369233, 1.752765374845802, 2.583172540894343, 3.566742989544581, 3.910575467696806, 4.629527666264413, 5.418493830140939, 5.846578122655711, 6.385783016248032, 7.231082823629955, 7.547742716759507, 8.238682347349025, 8.825044944647968, 9.389991156597378, 10.12306279808395, 10.49280660970112, 11.25994877268757, 11.48178544728753, 12.08357931802826, 12.55512061520299, 13.29662290243541, 13.64556874083923, 14.16766739367647, 14.84235144362981, 15.52581548975454