L(s) = 1 | − 3-s + 9-s − 6·13-s − 2·17-s + 8·19-s − 27-s − 2·29-s − 8·31-s − 37-s + 6·39-s + 10·41-s + 4·43-s + 4·47-s − 7·49-s + 2·51-s − 6·53-s − 8·57-s + 4·59-s + 2·61-s − 12·67-s + 8·71-s − 2·73-s − 8·79-s + 81-s − 4·83-s + 2·87-s + 10·89-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.164·37-s + 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.583·47-s − 49-s + 0.280·51-s − 0.824·53-s − 1.05·57-s + 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s + 0.214·87-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.81548236176068, −14.37217767445982, −14.07316043648524, −13.18199568186918, −12.85021918269737, −12.27766505328285, −11.84258628117376, −11.29050713090756, −10.89032376752245, −10.16211886995179, −9.661137429232537, −9.312273365007525, −8.720772062992455, −7.649505474747891, −7.482209619276428, −7.085593754270570, −6.181082092222408, −5.704934693543355, −5.049920438869072, −4.722113842996370, −3.908816881971299, −3.196519146746966, −2.481293849551964, −1.783053602967056, −0.8459205207675514, 0,
0.8459205207675514, 1.783053602967056, 2.481293849551964, 3.196519146746966, 3.908816881971299, 4.722113842996370, 5.049920438869072, 5.704934693543355, 6.181082092222408, 7.085593754270570, 7.482209619276428, 7.649505474747891, 8.720772062992455, 9.312273365007525, 9.661137429232537, 10.16211886995179, 10.89032376752245, 11.29050713090756, 11.84258628117376, 12.27766505328285, 12.85021918269737, 13.18199568186918, 14.07316043648524, 14.37217767445982, 14.81548236176068