| L(s) = 1 | − 16·4-s − 81·9-s − 120·11-s + 256·16-s − 1.91e3·19-s − 1.11e4·29-s − 7.18e3·31-s + 1.29e3·36-s + 3.83e4·41-s + 1.92e3·44-s + 2.63e3·49-s − 5.26e4·59-s − 6.21e4·61-s − 4.09e3·64-s + 1.22e4·71-s + 3.05e4·76-s − 1.48e5·79-s + 6.56e3·81-s + 6.54e4·89-s + 9.72e3·99-s − 4.40e4·101-s + 1.70e4·109-s + 1.78e5·116-s − 3.11e5·121-s + 1.14e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 0.299·11-s + 1/4·16-s − 1.21·19-s − 2.46·29-s − 1.34·31-s + 1/6·36-s + 3.56·41-s + 0.149·44-s + 0.156·49-s − 1.97·59-s − 2.13·61-s − 1/8·64-s + 0.288·71-s + 0.607·76-s − 2.68·79-s + 1/9·81-s + 0.876·89-s + 0.0996·99-s − 0.429·101-s + 0.137·109-s + 1.23·116-s − 1.93·121-s + 0.671·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4884239688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4884239688\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 2638 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 60 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 309622 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2668318 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 956 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 12512686 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5574 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3592 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 67150150 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 19194 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 116701030 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 71387614 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 141171770 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 26340 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 31090 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2417875798 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6120 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3492931822 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 74408 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7836246262 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 32742 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 10408550210 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.78000716418146201544645701105, −11.83128015831663460529586526303, −11.23891891189954213979634646636, −10.72053030259340718593677856013, −10.63116002478468723701034913328, −9.575084149249534643497780426903, −9.195786665547226751348102148403, −8.985090491676490937216647207068, −8.141401536547654964499188224041, −7.52992944312559253290244154219, −7.34167751211596913880753574596, −6.17061270592848741183560595170, −5.90207857256970746142712854777, −5.27540039637735631640722986797, −4.37200567852592493197811333331, −4.01463765050741375819661339958, −3.11671009215942532711482211859, −2.28695729710342799180328062468, −1.47035016809770809071589062797, −0.23627563398808766460666185056,
0.23627563398808766460666185056, 1.47035016809770809071589062797, 2.28695729710342799180328062468, 3.11671009215942532711482211859, 4.01463765050741375819661339958, 4.37200567852592493197811333331, 5.27540039637735631640722986797, 5.90207857256970746142712854777, 6.17061270592848741183560595170, 7.34167751211596913880753574596, 7.52992944312559253290244154219, 8.141401536547654964499188224041, 8.985090491676490937216647207068, 9.195786665547226751348102148403, 9.575084149249534643497780426903, 10.63116002478468723701034913328, 10.72053030259340718593677856013, 11.23891891189954213979634646636, 11.83128015831663460529586526303, 12.78000716418146201544645701105
