L(s) = 1 | − 3·3-s + 5-s − 7-s + 6·9-s + 11-s − 6·13-s − 3·15-s + 3·17-s + 5·19-s + 3·21-s + 2·23-s + 25-s − 9·27-s − 5·29-s − 5·31-s − 3·33-s − 35-s − 37-s + 18·39-s − 2·41-s − 12·43-s + 6·45-s + 2·47-s − 6·49-s − 9·51-s − 13·53-s + 55-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s − 0.377·7-s + 2·9-s + 0.301·11-s − 1.66·13-s − 0.774·15-s + 0.727·17-s + 1.14·19-s + 0.654·21-s + 0.417·23-s + 1/5·25-s − 1.73·27-s − 0.928·29-s − 0.898·31-s − 0.522·33-s − 0.169·35-s − 0.164·37-s + 2.88·39-s − 0.312·41-s − 1.82·43-s + 0.894·45-s + 0.291·47-s − 6/7·49-s − 1.26·51-s − 1.78·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.825680854346117849540954979854, −9.340325235669933130157390558235, −7.63099075673305127139126471709, −7.00785151235034331994887282351, −6.13093127977743725304890869032, −5.29137566483858541791858280760, −4.79526192988357236214226016972, −3.27987303917656365435943666175, −1.56608009407944446668478562756, 0,
1.56608009407944446668478562756, 3.27987303917656365435943666175, 4.79526192988357236214226016972, 5.29137566483858541791858280760, 6.13093127977743725304890869032, 7.00785151235034331994887282351, 7.63099075673305127139126471709, 9.340325235669933130157390558235, 9.825680854346117849540954979854