| L(s) = 1 | − 19.7·3-s − 10.8·5-s − 70.7·7-s + 146.·9-s − 539.·11-s + 766.·13-s + 213.·15-s − 703.·17-s − 276.·19-s + 1.39e3·21-s + 853.·23-s − 3.00e3·25-s + 1.89e3·27-s − 580.·29-s − 2.73e3·31-s + 1.06e4·33-s + 765.·35-s + 1.14e4·37-s − 1.51e4·39-s − 1.68e4·41-s + 1.75e4·43-s − 1.58e3·45-s + 2.31e4·47-s − 1.17e4·49-s + 1.38e4·51-s − 1.20e3·53-s + 5.83e3·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s − 0.193·5-s − 0.545·7-s + 0.604·9-s − 1.34·11-s + 1.25·13-s + 0.245·15-s − 0.590·17-s − 0.175·19-s + 0.691·21-s + 0.336·23-s − 0.962·25-s + 0.500·27-s − 0.128·29-s − 0.511·31-s + 1.70·33-s + 0.105·35-s + 1.37·37-s − 1.59·39-s − 1.56·41-s + 1.44·43-s − 0.116·45-s + 1.52·47-s − 0.701·49-s + 0.747·51-s − 0.0590·53-s + 0.260·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 19.7T + 243T^{2} \) |
| 5 | \( 1 + 10.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 70.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 539.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 766.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 703.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 276.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 853.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 580.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.75e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.20e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.55e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.61e4T + 8.58e9T^{2} \) |
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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
−8.775156259046441606521232488559, −7.903551925970701602284581910913, −6.92791520579206389507892020267, −6.05738235492862682920787744471, −5.55983651952653562770510115014, −4.56112655042121565456288661155, −3.52233238164850830473547536165, −2.31162036878885826030237240717, −0.836140443319889805482449907337, 0,
0.836140443319889805482449907337, 2.31162036878885826030237240717, 3.52233238164850830473547536165, 4.56112655042121565456288661155, 5.55983651952653562770510115014, 6.05738235492862682920787744471, 6.92791520579206389507892020267, 7.903551925970701602284581910913, 8.775156259046441606521232488559
