| L(s) = 1 | − 9·5-s + 7-s + 63·11-s + 28·13-s + 72·17-s + 98·19-s − 126·23-s − 44·25-s + 126·29-s + 259·31-s − 9·35-s − 386·37-s − 450·41-s − 34·43-s + 54·47-s − 342·49-s + 693·53-s − 567·55-s + 180·59-s + 280·61-s − 252·65-s − 586·67-s − 504·71-s + 161·73-s + 63·77-s − 440·79-s + 999·83-s + ⋯ |
| L(s) = 1 | − 0.804·5-s + 0.0539·7-s + 1.72·11-s + 0.597·13-s + 1.02·17-s + 1.18·19-s − 1.14·23-s − 0.351·25-s + 0.806·29-s + 1.50·31-s − 0.0434·35-s − 1.71·37-s − 1.71·41-s − 0.120·43-s + 0.167·47-s − 0.997·49-s + 1.79·53-s − 1.39·55-s + 0.397·59-s + 0.587·61-s − 0.480·65-s − 1.06·67-s − 0.842·71-s + 0.258·73-s + 0.0932·77-s − 0.626·79-s + 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.277530942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.277530942\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 - 63 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 98 T + p^{3} T^{2} \) |
| 23 | \( 1 + 126 T + p^{3} T^{2} \) |
| 29 | \( 1 - 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 259 T + p^{3} T^{2} \) |
| 37 | \( 1 + 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 450 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 54 T + p^{3} T^{2} \) |
| 53 | \( 1 - 693 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 280 T + p^{3} T^{2} \) |
| 67 | \( 1 + 586 T + p^{3} T^{2} \) |
| 71 | \( 1 + 504 T + p^{3} T^{2} \) |
| 73 | \( 1 - 161 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 - 999 T + p^{3} T^{2} \) |
| 89 | \( 1 - 882 T + p^{3} T^{2} \) |
| 97 | \( 1 + 721 T + p^{3} T^{2} \) |
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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
−8.774322866708180514698290668551, −8.269668268586204479698911522044, −7.37561441905678842848903624862, −6.61303566880615706873192824099, −5.81064400862075610908599113275, −4.73288074720000014386664143094, −3.76180754186193014466662989772, −3.31979307788105241443220062499, −1.66662083866779709062345744457, −0.76713892584382693098950697663,
0.76713892584382693098950697663, 1.66662083866779709062345744457, 3.31979307788105241443220062499, 3.76180754186193014466662989772, 4.73288074720000014386664143094, 5.81064400862075610908599113275, 6.61303566880615706873192824099, 7.37561441905678842848903624862, 8.269668268586204479698911522044, 8.774322866708180514698290668551
