| L(s) = 1 | − 8·25-s + 48·47-s − 2·49-s − 24·71-s + 40·73-s − 9·81-s + 8·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 8/5·25-s + 7.00·47-s − 2/7·49-s − 2.84·71-s + 4.68·73-s − 81-s + 0.812·97-s + 4·121-s + 0.0887·127-s + 0.0873·131-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 + 3.07·169-s + 0.0760·173-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 + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.447967591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447967591\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−6.95053400429764121749539875574, −6.69227441613206067540661816872, −6.26730928262159633670714046618, −6.25179397249292919492592124636, −5.92810601130871390578055574164, −5.79666651294002561796875171556, −5.48542527222416405323664276701, −5.36938608877597807671980636344, −5.34552400088163909847411344338, −4.77418507018176416283082275476, −4.54599955040675263023024731308, −4.25447359351113661426920067833, −4.18308477610880288272149520959, −3.88942931954054971697813552874, −3.83774917698097407002216912548, −3.30946589493670573847160165927, −3.08871469462771919111532343046, −2.89860909753782316651298128304, −2.46220368988041504503211971374, −2.19896121790962692168036762269, −1.90677320669866998330697495805, −1.90300953118654380244777770384, −0.984576197356245551386223654532, −0.905407052361729299583494746936, −0.45543606542336692927676059052,
0.45543606542336692927676059052, 0.905407052361729299583494746936, 0.984576197356245551386223654532, 1.90300953118654380244777770384, 1.90677320669866998330697495805, 2.19896121790962692168036762269, 2.46220368988041504503211971374, 2.89860909753782316651298128304, 3.08871469462771919111532343046, 3.30946589493670573847160165927, 3.83774917698097407002216912548, 3.88942931954054971697813552874, 4.18308477610880288272149520959, 4.25447359351113661426920067833, 4.54599955040675263023024731308, 4.77418507018176416283082275476, 5.34552400088163909847411344338, 5.36938608877597807671980636344, 5.48542527222416405323664276701, 5.79666651294002561796875171556, 5.92810601130871390578055574164, 6.25179397249292919492592124636, 6.26730928262159633670714046618, 6.69227441613206067540661816872, 6.95053400429764121749539875574
