| L(s) = 1 | − 8·7-s − 12·13-s − 8·25-s − 16·31-s − 12·37-s + 32·49-s + 16·67-s + 4·73-s + 96·91-s − 12·97-s + 8·103-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 64·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 3.32·13-s − 8/5·25-s − 2.87·31-s − 1.97·37-s + 32/7·49-s + 1.95·67-s + 0.468·73-s + 10.0·91-s − 1.21·97-s + 0.788·103-s + 2.54·121-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 + 5.53·169-s + 0.0760·173-s + 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1280350463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1280350463\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 254 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.40825693350748097855272874510, −7.21359377810603002845865786911, −7.14667016751687632627091982644, −6.70808549194727805996555403117, −6.48386041513352095327815771761, −6.42289560724105188641275513271, −6.40737718812415350190111408752, −5.55212560735151967607077628611, −5.50000292485725620623954091062, −5.48589798083742353203024673395, −5.34171549202662570852636868440, −4.94535050928679117811192896678, −4.48317662742546164836781586482, −4.21356399565083671514379527282, −4.03841521061702613344557650018, −3.64268188364035595652042383880, −3.44457019639751146849055941741, −3.13827739608353071299758350718, −3.03617743366233901509756974319, −2.44735317854314536331101603992, −2.38835102443905046876701905465, −1.84347149441485277805172706486, −1.81060055025570687130074595257, −0.56302754292156929325041158031, −0.15556259102759866391812967000,
0.15556259102759866391812967000, 0.56302754292156929325041158031, 1.81060055025570687130074595257, 1.84347149441485277805172706486, 2.38835102443905046876701905465, 2.44735317854314536331101603992, 3.03617743366233901509756974319, 3.13827739608353071299758350718, 3.44457019639751146849055941741, 3.64268188364035595652042383880, 4.03841521061702613344557650018, 4.21356399565083671514379527282, 4.48317662742546164836781586482, 4.94535050928679117811192896678, 5.34171549202662570852636868440, 5.48589798083742353203024673395, 5.50000292485725620623954091062, 5.55212560735151967607077628611, 6.40737718812415350190111408752, 6.42289560724105188641275513271, 6.48386041513352095327815771761, 6.70808549194727805996555403117, 7.14667016751687632627091982644, 7.21359377810603002845865786911, 7.40825693350748097855272874510
