| L(s) = 1 | + 2·2-s + 4·4-s + 8·8-s + 24·13-s + 16·16-s − 16·17-s + 48·26-s + 42·29-s + 32·32-s − 32·34-s − 24·37-s + 18·41-s + 49·49-s + 96·52-s + 56·53-s + 84·58-s + 22·61-s + 64·64-s − 64·68-s − 96·73-s − 48·74-s + 36·82-s − 78·89-s − 144·97-s + 98·98-s + 198·101-s + 192·104-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 8-s + 1.84·13-s + 16-s − 0.941·17-s + 1.84·26-s + 1.44·29-s + 32-s − 0.941·34-s − 0.648·37-s + 0.439·41-s + 49-s + 1.84·52-s + 1.05·53-s + 1.44·58-s + 0.360·61-s + 64-s − 0.941·68-s − 1.31·73-s − 0.648·74-s + 0.439·82-s − 0.876·89-s − 1.48·97-s + 98-s + 1.96·101-s + 1.84·104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.112692943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.112692943\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 24 T + p^{2} T^{2} \) |
| 17 | \( 1 + 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 24 T + p^{2} T^{2} \) |
| 41 | \( 1 - 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 96 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 78 T + p^{2} T^{2} \) |
| 97 | \( 1 + 144 T + p^{2} T^{2} \) |
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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
−10.26116382265854354736397354560, −8.900352383665828936983381608564, −8.234823633936812334721305932186, −7.05124954479606435802224374892, −6.34530245709741843604284805379, −5.57340092090265083949314338394, −4.44649069876883523749550840710, −3.68566606813130254385648025994, −2.56589198881453562956468087233, −1.24345267013438478459114313657,
1.24345267013438478459114313657, 2.56589198881453562956468087233, 3.68566606813130254385648025994, 4.44649069876883523749550840710, 5.57340092090265083949314338394, 6.34530245709741843604284805379, 7.05124954479606435802224374892, 8.234823633936812334721305932186, 8.900352383665828936983381608564, 10.26116382265854354736397354560
