| L(s) = 1 | − 6·3-s − 4·4-s + 27·9-s + 24·12-s + 78·13-s + 16·16-s − 92·17-s − 224·23-s + 186·25-s − 108·27-s − 340·29-s − 108·36-s − 468·39-s − 184·43-s − 96·48-s + 490·49-s + 552·51-s − 312·52-s + 1.11e3·53-s + 1.80e3·61-s − 64·64-s + 368·68-s + 1.34e3·69-s − 1.11e3·75-s + 720·79-s + 405·81-s + 2.04e3·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s − 1.31·17-s − 2.03·23-s + 1.48·25-s − 0.769·27-s − 2.17·29-s − 1/2·36-s − 1.92·39-s − 0.652·43-s − 0.288·48-s + 10/7·49-s + 1.51·51-s − 0.832·52-s + 2.89·53-s + 3.78·61-s − 1/8·64-s + 0.656·68-s + 2.34·69-s − 1.71·75-s + 1.02·79-s + 5/9·81-s + 2.51·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9219166616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9219166616\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( 1 - 6 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 186 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1762 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 9362 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 170 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 47482 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 101290 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6558 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 194650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 558 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 405282 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 902 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 184210 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 149078 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 85810 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1111890 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1368322 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 179710 T^{2} + p^{6} T^{4} \) |
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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
−14.03375486928543423498568936085, −13.52974561694248979938393825158, −13.09893146429623805640266099654, −12.74033134104201056748128376317, −11.85015084120389196718786971694, −11.54644167280272575073213266852, −10.93216030147573240115838160404, −10.48453570986699605643742743716, −9.928526958390818053313894247268, −9.048626315808177951026086334311, −8.660669249882331562723778521235, −7.954571734255209276626468126926, −6.95204020432829145106293551267, −6.55771327936691827416174948932, −5.62696273395737064085875951223, −5.39139994703095478308763637700, −4.02229882577256157610610015095, −3.95716147043557147030716167385, −2.05980032206304119942369128054, −0.67099761649555014309172669341,
0.67099761649555014309172669341, 2.05980032206304119942369128054, 3.95716147043557147030716167385, 4.02229882577256157610610015095, 5.39139994703095478308763637700, 5.62696273395737064085875951223, 6.55771327936691827416174948932, 6.95204020432829145106293551267, 7.954571734255209276626468126926, 8.660669249882331562723778521235, 9.048626315808177951026086334311, 9.928526958390818053313894247268, 10.48453570986699605643742743716, 10.93216030147573240115838160404, 11.54644167280272575073213266852, 11.85015084120389196718786971694, 12.74033134104201056748128376317, 13.09893146429623805640266099654, 13.52974561694248979938393825158, 14.03375486928543423498568936085
