| L(s) = 1 | + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 8·23-s − 6·29-s − 8·31-s + 2·37-s + 2·41-s − 12·43-s − 8·47-s + 6·53-s − 4·59-s + 2·61-s + 12·67-s + 8·71-s − 14·73-s + 12·83-s + 2·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s − 1.82·43-s − 1.16·47-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.63·73-s + 1.31·83-s + 0.211·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706254685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706254685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.22250772499897, −12.80355363496028, −12.25337633593195, −11.68392107821497, −11.30706053408288, −10.91901301809615, −10.41920844745713, −9.700196971764083, −9.378302567329046, −8.916463899286581, −8.534834111210672, −7.887755010234144, −7.253600138216777, −6.861924704243304, −6.511949136533669, −5.875815426505951, −5.231649842502073, −4.802333000208322, −4.240031017453571, −3.569918717847688, −3.285199667348784, −2.327984139989925, −1.898271404986577, −1.239706676228107, −0.3808751481519590,
0.3808751481519590, 1.239706676228107, 1.898271404986577, 2.327984139989925, 3.285199667348784, 3.569918717847688, 4.240031017453571, 4.802333000208322, 5.231649842502073, 5.875815426505951, 6.511949136533669, 6.861924704243304, 7.253600138216777, 7.887755010234144, 8.534834111210672, 8.916463899286581, 9.378302567329046, 9.700196971764083, 10.41920844745713, 10.91901301809615, 11.30706053408288, 11.68392107821497, 12.25337633593195, 12.80355363496028, 13.22250772499897
