| L(s) = 1 | − 2·2-s + 4·3-s + 2·4-s − 8·6-s − 4·8-s + 4·9-s + 8·12-s + 8·16-s − 8·18-s − 16·24-s − 4·27-s − 8·31-s − 8·32-s + 8·36-s + 8·37-s − 8·41-s − 28·43-s + 32·48-s + 20·49-s + 32·53-s + 8·54-s + 16·62-s + 8·64-s − 36·67-s + 8·71-s − 16·72-s − 16·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 4-s − 3.26·6-s − 1.41·8-s + 4/3·9-s + 2.30·12-s + 2·16-s − 1.88·18-s − 3.26·24-s − 0.769·27-s − 1.43·31-s − 1.41·32-s + 4/3·36-s + 1.31·37-s − 1.24·41-s − 4.26·43-s + 4.61·48-s + 20/7·49-s + 4.39·53-s + 1.08·54-s + 2.03·62-s + 64-s − 4.39·67-s + 0.949·71-s − 1.88·72-s − 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.164025938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164025938\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1274 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.121893544177709379473274233582, −8.611198104545520429956874499974, −8.574460175233462015115753246520, −8.545406457889532750429173903124, −8.396693322336625231350986220045, −7.989946772885703948650253364531, −7.56978981097575224447493628148, −7.50281848193954513407267736321, −7.10968471180960990171383998362, −6.93615623970837667152620247320, −6.37533703346812715166520916945, −6.32053956450899452943993228132, −5.69285397887016362610639577947, −5.65793535687404674032941093397, −5.08516966199271487860207398171, −5.00899091981260562276640246658, −4.15046156057961027481250194208, −3.95361207165823130152554861631, −3.54362985360426655564845850290, −3.11540259944749621306411411312, −3.02714902970405647398462517339, −2.62640780843944343045930744331, −2.00285850582407157288296123012, −1.91673788953325424389475291069, −0.72645289390318715026944689259,
0.72645289390318715026944689259, 1.91673788953325424389475291069, 2.00285850582407157288296123012, 2.62640780843944343045930744331, 3.02714902970405647398462517339, 3.11540259944749621306411411312, 3.54362985360426655564845850290, 3.95361207165823130152554861631, 4.15046156057961027481250194208, 5.00899091981260562276640246658, 5.08516966199271487860207398171, 5.65793535687404674032941093397, 5.69285397887016362610639577947, 6.32053956450899452943993228132, 6.37533703346812715166520916945, 6.93615623970837667152620247320, 7.10968471180960990171383998362, 7.50281848193954513407267736321, 7.56978981097575224447493628148, 7.989946772885703948650253364531, 8.396693322336625231350986220045, 8.545406457889532750429173903124, 8.574460175233462015115753246520, 8.611198104545520429956874499974, 9.121893544177709379473274233582
