| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 2·11-s − 12-s − 3·13-s + 14-s + 16-s − 4·17-s − 18-s + 21-s + 2·22-s − 4·23-s + 24-s + 3·26-s − 27-s − 28-s + 3·31-s − 32-s + 2·33-s + 4·34-s + 36-s − 5·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.218·21-s + 0.426·22-s − 0.834·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s − 0.188·28-s + 0.538·31-s − 0.176·32-s + 0.348·33-s + 0.685·34-s + 1/6·36-s − 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1699168309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1699168309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.55874774302939, −13.92159310260749, −13.22204400120306, −12.87328035575834, −12.33234039777566, −11.70811506038408, −11.42390067867226, −10.71134310661132, −10.29054048479360, −9.822720559471157, −9.407890197553641, −8.689334047455644, −8.137575734086896, −7.695512649901101, −6.853189625968154, −6.735120283689064, −5.999620513477236, −5.339980039541977, −4.856262235857813, −4.127785761131844, −3.413794257423387, −2.540430037272117, −2.135247985729873, −1.214682677805674, −0.1726841948440752,
0.1726841948440752, 1.214682677805674, 2.135247985729873, 2.540430037272117, 3.413794257423387, 4.127785761131844, 4.856262235857813, 5.339980039541977, 5.999620513477236, 6.735120283689064, 6.853189625968154, 7.695512649901101, 8.137575734086896, 8.689334047455644, 9.407890197553641, 9.822720559471157, 10.29054048479360, 10.71134310661132, 11.42390067867226, 11.70811506038408, 12.33234039777566, 12.87328035575834, 13.22204400120306, 13.92159310260749, 14.55874774302939
