| L(s) = 1 | − 4·7-s + 16·13-s − 16·19-s − 20·37-s − 28·43-s + 10·49-s + 16·61-s − 28·67-s − 64·91-s − 40·97-s + 16·103-s + 52·109-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4.43·13-s − 3.67·19-s − 3.28·37-s − 4.26·43-s + 10/7·49-s + 2.04·61-s − 3.42·67-s − 6.70·91-s − 4.06·97-s + 1.57·103-s + 4.98·109-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-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 + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.290952553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290952553\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1646 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 4046 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−5.90941006466283723664453798427, −5.86311682642044434272296251452, −5.86305698738240146533915991268, −5.33028258093053417328860863069, −5.23321677984293696802096041929, −5.14828387858008289339494625226, −4.55911885750188444370028787459, −4.50396442657349809579027122989, −4.25843238954037159629405639631, −4.18352806242583437148571878432, −3.98686072008818576098137874708, −3.42574705865315681205565319167, −3.42291226016467968963239975250, −3.41714937847170717130933719313, −3.32701044593420191966898336174, −3.14676676469055961882837740629, −2.51834235423621424738422275888, −2.35494525930556079659087333845, −1.93683646412438082399957755095, −1.80215997049526176719355365312, −1.62888437451601816455697276571, −1.40808762023501475524035457687, −1.02099167244626558750972894805, −0.44751470645164086790801955248, −0.23237381149250893244111173347,
0.23237381149250893244111173347, 0.44751470645164086790801955248, 1.02099167244626558750972894805, 1.40808762023501475524035457687, 1.62888437451601816455697276571, 1.80215997049526176719355365312, 1.93683646412438082399957755095, 2.35494525930556079659087333845, 2.51834235423621424738422275888, 3.14676676469055961882837740629, 3.32701044593420191966898336174, 3.41714937847170717130933719313, 3.42291226016467968963239975250, 3.42574705865315681205565319167, 3.98686072008818576098137874708, 4.18352806242583437148571878432, 4.25843238954037159629405639631, 4.50396442657349809579027122989, 4.55911885750188444370028787459, 5.14828387858008289339494625226, 5.23321677984293696802096041929, 5.33028258093053417328860863069, 5.86305698738240146533915991268, 5.86311682642044434272296251452, 5.90941006466283723664453798427
