| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 12·11-s + 12·12-s − 2·13-s − 14·14-s + 16·16-s + 18·17-s + 18·18-s + 56·19-s − 21·21-s + 24·22-s + 156·23-s + 24·24-s − 4·26-s + 27·27-s − 28·28-s − 186·29-s − 52·31-s + 32·32-s + 36·33-s + 36·34-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 + 0.328·11-s + 0.288·12-s − 0.0426·13-s − 0.267·14-s + 1/4·16-s + 0.256·17-s + 0.235·18-s + 0.676·19-s − 0.218·21-s + 0.232·22-s + 1.41·23-s + 0.204·24-s − 0.0301·26-s + 0.192·27-s − 0.188·28-s − 1.19·29-s − 0.301·31-s + 0.176·32-s + 0.189·33-s + 0.181·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.415965521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.415965521\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 - 156 T + p^{3} T^{2} \) |
| 29 | \( 1 + 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 52 T + p^{3} T^{2} \) |
| 37 | \( 1 - 178 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 456 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 936 T + p^{3} T^{2} \) |
| 73 | \( 1 + 542 T + p^{3} T^{2} \) |
| 79 | \( 1 - 992 T + p^{3} T^{2} \) |
| 83 | \( 1 - 276 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−9.416085751832697920953328611118, −8.855094747582629135594816948007, −7.60136480311271577110655231860, −7.12008257317066572905764541739, −6.03868043485663568949873487272, −5.18980710685609601827877094147, −4.09416343613532886810871876893, −3.29986400180229109725814716171, −2.36288461183529027670767179988, −1.01643576242145241486169525246,
1.01643576242145241486169525246, 2.36288461183529027670767179988, 3.29986400180229109725814716171, 4.09416343613532886810871876893, 5.18980710685609601827877094147, 6.03868043485663568949873487272, 7.12008257317066572905764541739, 7.60136480311271577110655231860, 8.855094747582629135594816948007, 9.416085751832697920953328611118
