| L(s) = 1 | − 4·2-s + 8·4-s + 18·5-s − 72·10-s + 50·11-s + 36·13-s − 64·16-s + 126·17-s + 72·19-s + 144·20-s − 200·22-s − 14·23-s + 199·25-s − 144·26-s − 158·29-s + 36·31-s + 256·32-s − 504·34-s − 162·37-s − 288·38-s − 270·41-s − 324·43-s + 400·44-s + 56·46-s − 72·47-s − 796·50-s + 288·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.60·5-s − 2.27·10-s + 1.37·11-s + 0.768·13-s − 16-s + 1.79·17-s + 0.869·19-s + 1.60·20-s − 1.93·22-s − 0.126·23-s + 1.59·25-s − 1.08·26-s − 1.01·29-s + 0.208·31-s + 1.41·32-s − 2.54·34-s − 0.719·37-s − 1.22·38-s − 1.02·41-s − 1.14·43-s + 1.37·44-s + 0.179·46-s − 0.223·47-s − 2.25·50-s + 0.768·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.546389654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.546389654\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 72 T + p^{3} T^{2} \) |
| 23 | \( 1 + 14 T + p^{3} T^{2} \) |
| 29 | \( 1 + 158 T + p^{3} T^{2} \) |
| 31 | \( 1 - 36 T + p^{3} T^{2} \) |
| 37 | \( 1 + 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 270 T + p^{3} T^{2} \) |
| 43 | \( 1 + 324 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 22 T + p^{3} T^{2} \) |
| 59 | \( 1 - 468 T + p^{3} T^{2} \) |
| 61 | \( 1 + 792 T + p^{3} T^{2} \) |
| 67 | \( 1 - 232 T + p^{3} T^{2} \) |
| 71 | \( 1 - 734 T + p^{3} T^{2} \) |
| 73 | \( 1 + 180 T + p^{3} T^{2} \) |
| 79 | \( 1 - 236 T + p^{3} T^{2} \) |
| 83 | \( 1 - 36 T + p^{3} T^{2} \) |
| 89 | \( 1 - 234 T + p^{3} T^{2} \) |
| 97 | \( 1 + 468 T + p^{3} T^{2} \) |
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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
−10.23033031976023580655049014979, −9.721459430642370501499305595259, −9.138265308090915139649259479095, −8.249119455653192645744076477598, −7.09547112697851912332151090138, −6.20830695344423707404086769149, −5.26215706401231760157048232762, −3.45862768474983275245374541917, −1.75354957020147024898294374465, −1.13271064717039472639225858843,
1.13271064717039472639225858843, 1.75354957020147024898294374465, 3.45862768474983275245374541917, 5.26215706401231760157048232762, 6.20830695344423707404086769149, 7.09547112697851912332151090138, 8.249119455653192645744076477598, 9.138265308090915139649259479095, 9.721459430642370501499305595259, 10.23033031976023580655049014979
