L(s) = 1 | − 2·5-s + 7-s + 6·11-s − 4·17-s − 19-s + 4·23-s − 25-s − 2·29-s − 2·31-s − 2·35-s − 10·41-s − 4·43-s + 4·47-s + 49-s − 14·53-s − 12·55-s − 2·61-s + 12·67-s − 2·73-s + 6·77-s + 10·79-s − 6·83-s + 8·85-s − 10·89-s + 2·95-s − 2·97-s − 6·101-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s + 1.80·11-s − 0.970·17-s − 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.338·35-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.92·53-s − 1.61·55-s − 0.256·61-s + 1.46·67-s − 0.234·73-s + 0.683·77-s + 1.12·79-s − 0.658·83-s + 0.867·85-s − 1.05·89-s + 0.205·95-s − 0.203·97-s − 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.22691893437655949048325903421, −6.76790358391537788637589702705, −6.16470799052164746476181404201, −5.13367461976556064774893175522, −4.45481007992373788200109568817, −3.85516632097326418693035259867, −3.26413020570320937120106891489, −2.03516272748559446734667846717, −1.24913849907889685446558554611, 0,
1.24913849907889685446558554611, 2.03516272748559446734667846717, 3.26413020570320937120106891489, 3.85516632097326418693035259867, 4.45481007992373788200109568817, 5.13367461976556064774893175522, 6.16470799052164746476181404201, 6.76790358391537788637589702705, 7.22691893437655949048325903421