| L(s) = 1 | − 8·11-s + 4·17-s + 8·19-s − 2·25-s + 20·41-s + 24·43-s − 6·49-s + 8·59-s + 8·67-s + 4·73-s + 8·83-s + 12·89-s + 28·97-s − 24·107-s − 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 0.970·17-s + 1.83·19-s − 2/5·25-s + 3.12·41-s + 3.65·43-s − 6/7·49-s + 1.04·59-s + 0.977·67-s + 0.468·73-s + 0.878·83-s + 1.27·89-s + 2.84·97-s − 2.32·107-s − 0.376·113-s + 2.36·121-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 + 6/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564528418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564528418\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.327795282091750540174519367152, −8.894924557433946970025276376494, −8.202722803732038260819914072311, −7.938448208337795863836328320566, −7.56937116800425618663175947393, −7.54298636928153294021071163057, −7.15373393611909754800850421984, −6.31560238075764798982347559811, −5.99165327885955224849963000095, −5.52856038211060994272085134467, −5.34922354924852502163277944794, −5.02403853152553644520129139008, −4.29233427240940603052381572951, −4.05054195852994500497621289189, −3.18788431145204240549562376929, −3.07802410675241370947638387044, −2.29150084540214338026025771785, −2.27750845644887222463745971397, −0.947900956292784675730688449337, −0.70935874290541488313300526075,
0.70935874290541488313300526075, 0.947900956292784675730688449337, 2.27750845644887222463745971397, 2.29150084540214338026025771785, 3.07802410675241370947638387044, 3.18788431145204240549562376929, 4.05054195852994500497621289189, 4.29233427240940603052381572951, 5.02403853152553644520129139008, 5.34922354924852502163277944794, 5.52856038211060994272085134467, 5.99165327885955224849963000095, 6.31560238075764798982347559811, 7.15373393611909754800850421984, 7.54298636928153294021071163057, 7.56937116800425618663175947393, 7.938448208337795863836328320566, 8.202722803732038260819914072311, 8.894924557433946970025276376494, 9.327795282091750540174519367152
