| L(s) = 1 | − 4·3-s − 6·5-s + 27·9-s + 12·11-s − 164·13-s + 24·15-s + 30·17-s − 68·19-s − 216·23-s + 125·25-s − 260·27-s + 492·29-s + 112·31-s − 48·33-s − 110·37-s + 656·39-s − 492·41-s − 344·43-s − 162·45-s − 192·47-s − 120·51-s − 558·53-s − 72·55-s + 272·57-s − 540·59-s − 110·61-s + 984·65-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.536·5-s + 9-s + 0.328·11-s − 3.49·13-s + 0.413·15-s + 0.428·17-s − 0.821·19-s − 1.95·23-s + 25-s − 1.85·27-s + 3.15·29-s + 0.648·31-s − 0.253·33-s − 0.488·37-s + 2.69·39-s − 1.87·41-s − 1.21·43-s − 0.536·45-s − 0.595·47-s − 0.329·51-s − 1.44·53-s − 0.176·55-s + 0.632·57-s − 1.19·59-s − 0.230·61-s + 1.87·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2978086248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2978086248\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T - 11 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T - 1187 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 82 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 30 T - 4013 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 68 T - 2235 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 216 T + 34489 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 112 T - 17247 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 192 T - 66959 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 558 T + 162487 T^{2} + 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 540 T + 86221 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 110 T - 214881 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 140 T - 281163 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 550 T - 86517 T^{2} - 550 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 208 T - 449775 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 516 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1398 T + 1249435 T^{2} - 1398 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1586 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10990275456284331313979778812, −11.88053978802689795514574928802, −11.79188901751146939980168353871, −10.54989104233363636548041475150, −10.30662685768885459774046504083, −9.987021935702769805201489248938, −9.595615729319619324015713385917, −8.783763886731685448797685357591, −8.017871226687383828220941146861, −7.72392620067272347833698178259, −7.17032583369822417280083927410, −6.45917709767366171036123246480, −6.31931685194823523962653104055, −4.93372762172154909881859677125, −4.91866246350272275700469118060, −4.45366831255890056571636164624, −3.39910665228630714643905484122, −2.52773375524641331173771060948, −1.68793326661785399715810453616, −0.24248465276623332536090872727,
0.24248465276623332536090872727, 1.68793326661785399715810453616, 2.52773375524641331173771060948, 3.39910665228630714643905484122, 4.45366831255890056571636164624, 4.91866246350272275700469118060, 4.93372762172154909881859677125, 6.31931685194823523962653104055, 6.45917709767366171036123246480, 7.17032583369822417280083927410, 7.72392620067272347833698178259, 8.017871226687383828220941146861, 8.783763886731685448797685357591, 9.595615729319619324015713385917, 9.987021935702769805201489248938, 10.30662685768885459774046504083, 10.54989104233363636548041475150, 11.79188901751146939980168353871, 11.88053978802689795514574928802, 12.10990275456284331313979778812
