| L(s) = 1 | − 4·4-s − 9·9-s − 144·11-s + 16·16-s − 184·19-s + 228·29-s + 112·31-s + 36·36-s + 12·41-s + 576·44-s − 49·49-s + 984·59-s − 500·61-s − 64·64-s + 72·71-s + 736·76-s − 112·79-s + 81·81-s − 780·89-s + 1.29e3·99-s − 2.70e3·101-s + 2.22e3·109-s − 912·116-s + 1.28e4·121-s − 448·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 3.94·11-s + 1/4·16-s − 2.22·19-s + 1.45·29-s + 0.648·31-s + 1/6·36-s + 0.0457·41-s + 1.97·44-s − 1/7·49-s + 2.17·59-s − 1.04·61-s − 1/8·64-s + 0.120·71-s + 1.11·76-s − 0.159·79-s + 1/9·81-s − 0.928·89-s + 1.31·99-s − 2.66·101-s + 1.95·109-s − 0.729·116-s + 9.68·121-s − 0.324·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1027508696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1027508696\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9790 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8066 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 114 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 132118 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 179422 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 129962 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 492 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 586150 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 242066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1091590 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 390 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1820446 T^{2} + p^{6} T^{4} \) |
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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
−9.902572049413084105423211540504, −9.427068687014456601852482409341, −8.512656660960623474220395965697, −8.491964011067177279315270661250, −8.182550628865246096383110765188, −7.930031724357838386746408102899, −7.21373118182298356981701827252, −6.97030323722475017830532386355, −6.15181513939298232046157140951, −5.91849451028325975658258516513, −5.21577304542643801661899534120, −5.16973699601841781779310227131, −4.44718156564669562171139120473, −4.31735279856206204945564502214, −3.26368182305083798240671594322, −2.87732912128440471104072197174, −2.36795900121303239473496621988, −2.10414078823841281204173772513, −0.834321493378490816112253835002, −0.099657920157259991319029215947,
0.099657920157259991319029215947, 0.834321493378490816112253835002, 2.10414078823841281204173772513, 2.36795900121303239473496621988, 2.87732912128440471104072197174, 3.26368182305083798240671594322, 4.31735279856206204945564502214, 4.44718156564669562171139120473, 5.16973699601841781779310227131, 5.21577304542643801661899534120, 5.91849451028325975658258516513, 6.15181513939298232046157140951, 6.97030323722475017830532386355, 7.21373118182298356981701827252, 7.930031724357838386746408102899, 8.182550628865246096383110765188, 8.491964011067177279315270661250, 8.512656660960623474220395965697, 9.427068687014456601852482409341, 9.902572049413084105423211540504
