| L(s) = 1 | + 12·7-s + 4·9-s + 12·23-s + 24·31-s + 24·41-s − 12·47-s + 68·49-s + 48·63-s − 24·71-s + 16·73-s + 48·79-s + 6·81-s − 24·89-s − 16·97-s + 12·103-s + 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 4.53·7-s + 4/3·9-s + 2.50·23-s + 4.31·31-s + 3.74·41-s − 1.75·47-s + 68/7·49-s + 6.04·63-s − 2.84·71-s + 1.87·73-s + 5.40·79-s + 2/3·81-s − 2.54·89-s − 1.62·97-s + 1.18·103-s + 2.25·113-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 + 11.3·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(17.13308183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.13308183\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.87408489466295983647596598392, −6.43599974479509101738743143551, −6.32609026687056091261396070132, −6.03671401963567622367475751265, −5.91537943091034196527328501883, −5.37370544911769546755863656317, −5.22193474273691020405312855045, −5.08293514593266348123808184123, −4.95285992577437578539031471022, −4.65014674744689760221769797152, −4.51210613575953620226129260921, −4.50777750953115654925890762381, −4.24573980943864695091015876783, −4.02682533207793405846329703041, −3.57360831731957384378284412077, −3.28134458687571959580922266287, −2.87733466202939729634501969323, −2.63921646402657013464883227086, −2.38341900110084521354369894903, −2.12734496613596434979142475954, −1.82388916887065439124114505538, −1.41889562318520764193328000033, −1.06877578716346748447680640523, −1.04103282447108542597421830449, −0.927734229515040444581433338522,
0.927734229515040444581433338522, 1.04103282447108542597421830449, 1.06877578716346748447680640523, 1.41889562318520764193328000033, 1.82388916887065439124114505538, 2.12734496613596434979142475954, 2.38341900110084521354369894903, 2.63921646402657013464883227086, 2.87733466202939729634501969323, 3.28134458687571959580922266287, 3.57360831731957384378284412077, 4.02682533207793405846329703041, 4.24573980943864695091015876783, 4.50777750953115654925890762381, 4.51210613575953620226129260921, 4.65014674744689760221769797152, 4.95285992577437578539031471022, 5.08293514593266348123808184123, 5.22193474273691020405312855045, 5.37370544911769546755863656317, 5.91537943091034196527328501883, 6.03671401963567622367475751265, 6.32609026687056091261396070132, 6.43599974479509101738743143551, 6.87408489466295983647596598392
