| L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s − 2·11-s − 13-s + 2·15-s + 17-s − 2·19-s + 4·21-s + 8·23-s − 25-s + 27-s + 2·29-s − 4·31-s − 2·33-s + 8·35-s + 8·37-s − 39-s − 10·41-s + 2·45-s + 8·47-s + 9·49-s + 51-s − 2·53-s − 4·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.458·19-s + 0.872·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.348·33-s + 1.35·35-s + 1.31·37-s − 0.160·39-s − 1.56·41-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.140·51-s − 0.274·53-s − 0.539·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.070579555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.070579555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.69399460950926, −15.92510910343638, −15.08142001758183, −14.87696132168192, −14.33004156074182, −13.71162358151637, −13.22442744166091, −12.72179806461006, −11.87135367894897, −11.28122395654848, −10.63471584288408, −10.19694615582503, −9.395097915084181, −8.873593748935786, −8.249571057472508, −7.675566955683405, −7.098645096824280, −6.235879324759636, −5.361940539231294, −4.993923917838953, −4.271589756166169, −3.247302397357388, −2.380312787102766, −1.870308439624586, −0.9590617211748903,
0.9590617211748903, 1.870308439624586, 2.380312787102766, 3.247302397357388, 4.271589756166169, 4.993923917838953, 5.361940539231294, 6.235879324759636, 7.098645096824280, 7.675566955683405, 8.249571057472508, 8.873593748935786, 9.395097915084181, 10.19694615582503, 10.63471584288408, 11.28122395654848, 11.87135367894897, 12.72179806461006, 13.22442744166091, 13.71162358151637, 14.33004156074182, 14.87696132168192, 15.08142001758183, 15.92510910343638, 16.69399460950926
