| L(s) = 1 | − 3-s + 4-s + 8·7-s + 3·9-s − 12-s + 16-s − 6·17-s − 8·21-s − 8·27-s + 8·28-s + 3·36-s + 4·37-s − 24·41-s + 4·43-s + 18·47-s − 48-s + 34·49-s + 6·51-s − 12·59-s + 24·63-s + 5·64-s + 4·67-s − 6·68-s + 2·79-s + 8·81-s + 24·83-s − 8·84-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 3.02·7-s + 9-s − 0.288·12-s + 1/4·16-s − 1.45·17-s − 1.74·21-s − 1.53·27-s + 1.51·28-s + 1/2·36-s + 0.657·37-s − 3.74·41-s + 0.609·43-s + 2.62·47-s − 0.144·48-s + 34/7·49-s + 0.840·51-s − 1.56·59-s + 3.02·63-s + 5/8·64-s + 0.488·67-s − 0.727·68-s + 0.225·79-s + 8/9·81-s + 2.63·83-s − 0.872·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.640368617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.640368617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - T^{2} + 264 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 97 T^{2} + 3960 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4134 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - T + 150 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 265 T^{2} + 32736 T^{4} - 265 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.86285366798600350287657257982, −7.71404109989980366446907831645, −7.33661346892967501029086713857, −7.12089671327371578529796418527, −7.07909835709139186926625826554, −6.41934550863339423358096064731, −6.41433273282958980863982395942, −6.39149014839391517138188717269, −5.73381317262294266287031793152, −5.54141190815059891047079417856, −5.41633828393702255261623700285, −4.95523922174284795939583895398, −4.75464152156747080817414625846, −4.66036025455502599878801756939, −4.57903511795801412383980605138, −3.96551574810931030896192406226, −3.77914869807136625027393278089, −3.58518614421039986683783069661, −3.08884870290844587796562812934, −2.25554641029650655652788535509, −2.23246566136237335555898934150, −2.15558278360804170552666146541, −1.45756697331492100396381759157, −1.40143745429362552810299998003, −0.64796113728359139994112677424,
0.64796113728359139994112677424, 1.40143745429362552810299998003, 1.45756697331492100396381759157, 2.15558278360804170552666146541, 2.23246566136237335555898934150, 2.25554641029650655652788535509, 3.08884870290844587796562812934, 3.58518614421039986683783069661, 3.77914869807136625027393278089, 3.96551574810931030896192406226, 4.57903511795801412383980605138, 4.66036025455502599878801756939, 4.75464152156747080817414625846, 4.95523922174284795939583895398, 5.41633828393702255261623700285, 5.54141190815059891047079417856, 5.73381317262294266287031793152, 6.39149014839391517138188717269, 6.41433273282958980863982395942, 6.41934550863339423358096064731, 7.07909835709139186926625826554, 7.12089671327371578529796418527, 7.33661346892967501029086713857, 7.71404109989980366446907831645, 7.86285366798600350287657257982
