| L(s) = 1 | − 4·4-s − 24·7-s + 12·13-s + 12·16-s + 48·19-s + 16·25-s + 96·28-s − 176·31-s − 44·37-s − 72·43-s + 188·49-s − 48·52-s − 192·61-s − 32·64-s + 40·67-s + 312·73-s − 192·76-s + 240·79-s − 288·91-s + 128·97-s − 64·100-s − 88·103-s − 108·109-s − 288·112-s + 704·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 3.42·7-s + 0.923·13-s + 3/4·16-s + 2.52·19-s + 0.639·25-s + 24/7·28-s − 5.67·31-s − 1.18·37-s − 1.67·43-s + 3.83·49-s − 0.923·52-s − 3.14·61-s − 1/2·64-s + 0.597·67-s + 4.27·73-s − 2.52·76-s + 3.03·79-s − 3.16·91-s + 1.31·97-s − 0.639·100-s − 0.854·103-s − 0.990·109-s − 2.57·112-s + 5.67·124-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3026370163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3026370163\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 639 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 12 T + 122 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 400 T^{2} + 152367 T^{4} - 400 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 24 T + 674 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 236 T^{2} + 463014 T^{4} + 236 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2896 T^{2} + 3499359 T^{4} - 2896 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 22 T + 1131 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2800 T^{2} + 5418447 T^{4} - 2800 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 36 T + 3914 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1348 T^{2} + 7165446 T^{4} - 1348 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1264 T^{2} + 4737759 T^{4} - 1264 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 340 T^{2} + 8338374 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 96 T + 4946 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 20 T + 8106 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1418 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 156 T + 16550 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 120 T + 13010 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 25252 T^{2} + 253337190 T^{4} - 25252 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 17776 T^{2} + 184927599 T^{4} - 17776 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 64 T + 10095 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.33739785608368894079740858176, −5.83918851513588369918835830154, −5.80024530882516888119100718936, −5.75886772349010288142227210715, −5.45775882861767247434659501926, −5.23495325592525958992505164279, −5.04097486426203849921667858321, −4.88004885749470360762375039438, −4.58830086377978879529570831174, −4.29940368565659487276035079714, −3.74562302154522316931599093395, −3.67476696576976232772487751847, −3.63709204645483839547447323377, −3.36463439229246753556376248705, −3.22304473063388056689154253362, −3.13380249965255664744249137364, −3.12570788933997107972460935749, −2.36593146018583025824301397360, −1.99980621547069154477875539513, −1.86437383348509478878579433304, −1.63436583449688292024468224835, −1.02640863277663094907866887234, −0.825065159622911320431366221808, −0.37145762863690983610996238881, −0.13718705575798201582053835575,
0.13718705575798201582053835575, 0.37145762863690983610996238881, 0.825065159622911320431366221808, 1.02640863277663094907866887234, 1.63436583449688292024468224835, 1.86437383348509478878579433304, 1.99980621547069154477875539513, 2.36593146018583025824301397360, 3.12570788933997107972460935749, 3.13380249965255664744249137364, 3.22304473063388056689154253362, 3.36463439229246753556376248705, 3.63709204645483839547447323377, 3.67476696576976232772487751847, 3.74562302154522316931599093395, 4.29940368565659487276035079714, 4.58830086377978879529570831174, 4.88004885749470360762375039438, 5.04097486426203849921667858321, 5.23495325592525958992505164279, 5.45775882861767247434659501926, 5.75886772349010288142227210715, 5.80024530882516888119100718936, 5.83918851513588369918835830154, 6.33739785608368894079740858176
