| L(s) = 1 | + 8·7-s + 12·19-s + 16·25-s − 16·43-s + 12·49-s + 32·61-s + 24·73-s + 40·121-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 128·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 2.75·19-s + 16/5·25-s − 2.43·43-s + 12/7·49-s + 4.09·61-s + 2.80·73-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 9.67·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.40785186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.40785186\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.21833839051215522768527217794, −6.10381519300904640466639238464, −5.74101773079112090858776722990, −5.38769261356314720497995424846, −5.31209066896230370925768721154, −5.02603679095772538397602354344, −5.00856520536625212872518152796, −5.00220754565280265466849498750, −4.88977620412708082413290471036, −4.47942923083300307876074944368, −4.17701121121912962874466732426, −4.04591527062066017000191404609, −3.71451424182130498543812970706, −3.39305720260309779602237103757, −3.12483124892572180294816659802, −3.09018964519109165233542858563, −2.96168702141888287387910092955, −2.30105106626850360971507890753, −2.22593475380799848415361888597, −1.91539942357036517008013627298, −1.61939724843390129630547279697, −1.45092584749131381905973905005, −0.968845206439867045990154352915, −0.913453083315965222952609343829, −0.58761729278060226075713295127,
0.58761729278060226075713295127, 0.913453083315965222952609343829, 0.968845206439867045990154352915, 1.45092584749131381905973905005, 1.61939724843390129630547279697, 1.91539942357036517008013627298, 2.22593475380799848415361888597, 2.30105106626850360971507890753, 2.96168702141888287387910092955, 3.09018964519109165233542858563, 3.12483124892572180294816659802, 3.39305720260309779602237103757, 3.71451424182130498543812970706, 4.04591527062066017000191404609, 4.17701121121912962874466732426, 4.47942923083300307876074944368, 4.88977620412708082413290471036, 5.00220754565280265466849498750, 5.00856520536625212872518152796, 5.02603679095772538397602354344, 5.31209066896230370925768721154, 5.38769261356314720497995424846, 5.74101773079112090858776722990, 6.10381519300904640466639238464, 6.21833839051215522768527217794
