| L(s) = 1 | + 3·3-s + 3·5-s − 7-s + 6·9-s − 6·13-s + 9·15-s − 17-s − 8·19-s − 3·21-s − 4·23-s + 4·25-s + 9·27-s + 2·29-s − 11·31-s − 3·35-s + 8·37-s − 18·39-s + 2·41-s − 9·43-s + 18·45-s + 8·47-s + 49-s − 3·51-s + 5·53-s − 24·57-s + 4·59-s − 61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.377·7-s + 2·9-s − 1.66·13-s + 2.32·15-s − 0.242·17-s − 1.83·19-s − 0.654·21-s − 0.834·23-s + 4/5·25-s + 1.73·27-s + 0.371·29-s − 1.97·31-s − 0.507·35-s + 1.31·37-s − 2.88·39-s + 0.312·41-s − 1.37·43-s + 2.68·45-s + 1.16·47-s + 1/7·49-s − 0.420·51-s + 0.686·53-s − 3.17·57-s + 0.520·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.410152537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.410152537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.05812426432986, −12.64643683811679, −12.35113293010418, −11.58822035200879, −10.81566208592989, −10.29968524669156, −10.05170125850907, −9.574088933187536, −9.098838841672180, −8.982691867048461, −8.204390635135379, −7.864646885284480, −7.295966622382064, −6.737489903853144, −6.441960200825277, −5.639042792501834, −5.280824241258302, −4.374674320658573, −4.179893548261481, −3.483965804122245, −2.709482655903558, −2.431498237745024, −2.002717738237102, −1.638375196729118, −0.4488545920368715,
0.4488545920368715, 1.638375196729118, 2.002717738237102, 2.431498237745024, 2.709482655903558, 3.483965804122245, 4.179893548261481, 4.374674320658573, 5.280824241258302, 5.639042792501834, 6.441960200825277, 6.737489903853144, 7.295966622382064, 7.864646885284480, 8.204390635135379, 8.982691867048461, 9.098838841672180, 9.574088933187536, 10.05170125850907, 10.29968524669156, 10.81566208592989, 11.58822035200879, 12.35113293010418, 12.64643683811679, 13.05812426432986
