| L(s) = 1 | + 3·3-s − 6·5-s − 10·7-s + 6·9-s − 33·13-s − 18·15-s − 6·17-s − 26·19-s − 30·21-s + 24·23-s + 25·25-s + 9·27-s + 54·29-s + 60·35-s − 99·39-s − 72·41-s + 25·43-s − 36·45-s + 42·47-s − 23·49-s − 18·51-s + 108·53-s − 78·57-s + 126·59-s − 43·61-s − 60·63-s + 198·65-s + ⋯ |
| L(s) = 1 | + 3-s − 6/5·5-s − 1.42·7-s + 2/3·9-s − 2.53·13-s − 6/5·15-s − 0.352·17-s − 1.36·19-s − 1.42·21-s + 1.04·23-s + 25-s + 1/3·27-s + 1.86·29-s + 12/7·35-s − 2.53·39-s − 1.75·41-s + 0.581·43-s − 4/5·45-s + 0.893·47-s − 0.469·49-s − 0.352·51-s + 2.03·53-s − 1.36·57-s + 2.13·59-s − 0.704·61-s − 0.952·63-s + 3.04·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9919321989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9919321989\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 26 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 33 T + 532 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T - 253 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 24 T + 47 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T + 1813 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1055 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 937 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 72 T + 3409 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 25 T - 1224 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 42 T - 445 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 108 T + 6697 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 126 T + 8773 T^{2} - 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 43 T - 1872 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 99 T + 7756 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 108 T + 8929 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T - 5208 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T + 6244 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 8029 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 228 T + 26737 T^{2} + 228 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.28039520899868148901631670271, −12.04667645303336359855453112082, −11.35274065887941482412674397177, −10.52708160680471580436802026755, −10.35172240789053762087231109445, −9.563891797235823217326973317519, −9.533783035508115559390341115025, −8.595558167358451664338696098152, −8.460003288990816618819398003615, −7.78041482425184826043155242191, −7.18821231272517340933532809387, −6.67322165885592258227476517847, −6.63603752908712849133478371078, −5.19855145135048883164164192667, −4.82604878365279261810696711862, −4.03791869702501604690629437023, −3.54883228094232497437087847175, −2.57060394381852747567525188698, −2.52557850793510552896131464964, −0.48832807278622459681448352052,
0.48832807278622459681448352052, 2.52557850793510552896131464964, 2.57060394381852747567525188698, 3.54883228094232497437087847175, 4.03791869702501604690629437023, 4.82604878365279261810696711862, 5.19855145135048883164164192667, 6.63603752908712849133478371078, 6.67322165885592258227476517847, 7.18821231272517340933532809387, 7.78041482425184826043155242191, 8.460003288990816618819398003615, 8.595558167358451664338696098152, 9.533783035508115559390341115025, 9.563891797235823217326973317519, 10.35172240789053762087231109445, 10.52708160680471580436802026755, 11.35274065887941482412674397177, 12.04667645303336359855453112082, 12.28039520899868148901631670271
