| L(s) = 1 | + 6·9-s − 16·13-s − 2·25-s + 16·37-s + 4·49-s + 40·61-s − 8·73-s + 27·81-s + 40·97-s − 40·109-s − 96·117-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·9-s − 4.43·13-s − 2/5·25-s + 2.63·37-s + 4/7·49-s + 5.12·61-s − 0.936·73-s + 3·81-s + 4.06·97-s − 3.83·109-s − 8.87·117-s − 1.81·121-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 + 8.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.676076429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.676076429\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−7.11497971684968336468316248350, −6.99383880967085274439688156584, −6.82580653174917820999052156720, −6.69837545135997129217770093922, −6.50209597120957223609352888581, −5.91290086632325466704977949110, −5.81992161310353618861476197595, −5.59029648855747847446993202119, −5.11488404002943330331590815223, −5.01015137904076881958405818697, −5.00500413134931241950736903349, −4.62727704328125151185581494693, −4.30505036211368661504363219478, −4.27254604993901083543535084823, −3.89865217245323185442279801352, −3.75193665852114912527313561294, −3.29956783434892709884503642743, −2.74519630418242092018695059058, −2.66395661926230098667515205010, −2.41587507820056681957513959122, −2.06121599581559811849482216811, −2.00009676375686446543975626198, −1.35594937719992004067689183599, −0.76714289168996611660210428909, −0.48937519566131699730116159891,
0.48937519566131699730116159891, 0.76714289168996611660210428909, 1.35594937719992004067689183599, 2.00009676375686446543975626198, 2.06121599581559811849482216811, 2.41587507820056681957513959122, 2.66395661926230098667515205010, 2.74519630418242092018695059058, 3.29956783434892709884503642743, 3.75193665852114912527313561294, 3.89865217245323185442279801352, 4.27254604993901083543535084823, 4.30505036211368661504363219478, 4.62727704328125151185581494693, 5.00500413134931241950736903349, 5.01015137904076881958405818697, 5.11488404002943330331590815223, 5.59029648855747847446993202119, 5.81992161310353618861476197595, 5.91290086632325466704977949110, 6.50209597120957223609352888581, 6.69837545135997129217770093922, 6.82580653174917820999052156720, 6.99383880967085274439688156584, 7.11497971684968336468316248350
