| L(s) = 1 | − 2·2-s + 2·3-s + 4-s + 2·5-s − 4·6-s + 2·8-s + 9-s − 4·10-s − 4·11-s + 2·12-s + 4·15-s − 4·16-s − 2·18-s + 2·20-s + 8·22-s − 4·23-s + 4·24-s + 25-s − 2·27-s + 16·29-s − 8·30-s − 4·31-s + 2·32-s − 8·33-s + 36-s + 4·40-s + 24·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s + 0.707·8-s + 1/3·9-s − 1.26·10-s − 1.20·11-s + 0.577·12-s + 1.03·15-s − 16-s − 0.471·18-s + 0.447·20-s + 1.70·22-s − 0.834·23-s + 0.816·24-s + 1/5·25-s − 0.384·27-s + 2.97·29-s − 1.46·30-s − 0.718·31-s + 0.353·32-s − 1.39·33-s + 1/6·36-s + 0.632·40-s + 3.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.882239447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.882239447\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 184 T^{3} - 1201 T^{4} - 184 p T^{5} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 72 T^{2} + 3815 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 120 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 32 T^{2} - 184 T^{3} - 657 T^{4} - 184 p T^{5} - 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 496 T^{3} - 3333 T^{4} - 496 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 62 T^{2} - 432 T^{3} - 5253 T^{4} - 432 p T^{5} + 62 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 18 T^{2} + 48 T^{3} + 3371 T^{4} + 48 p T^{5} + 18 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 24 T^{2} + 376 T^{3} - 3961 T^{4} + 376 p T^{5} - 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 102 T^{2} + 112 T^{3} + 7427 T^{4} + 112 p T^{5} - 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 30 T^{2} - 5341 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 94 T^{2} + 272 T^{3} + 2755 T^{4} + 272 p T^{5} - 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.88467179314056365041275759683, −6.47493504717400522196668687869, −6.36220492138492771204754876556, −6.24379037902097920300892982671, −6.04838804013094700368736240085, −5.72044241218869125995973502928, −5.45568883557095004276687322825, −5.36919440661996387875593296889, −4.97797299996805279536919651508, −4.76387889411476678885022073650, −4.54036799671397139120768088094, −4.33942250385754095274347934415, −3.96594148434104622741039151844, −3.94977606435075788689933479281, −3.43270072598801718893289850816, −3.25413934576986458118547600675, −2.79852408529432256747290262419, −2.67844886011631871232655950755, −2.35356893214594785948174154581, −2.22719185993947789928075511618, −2.16471505500199443364508523404, −1.39027138690386347039920585880, −1.24186694914375611279756414302, −0.69896090537668797638094349520, −0.52611030048546418460169902068,
0.52611030048546418460169902068, 0.69896090537668797638094349520, 1.24186694914375611279756414302, 1.39027138690386347039920585880, 2.16471505500199443364508523404, 2.22719185993947789928075511618, 2.35356893214594785948174154581, 2.67844886011631871232655950755, 2.79852408529432256747290262419, 3.25413934576986458118547600675, 3.43270072598801718893289850816, 3.94977606435075788689933479281, 3.96594148434104622741039151844, 4.33942250385754095274347934415, 4.54036799671397139120768088094, 4.76387889411476678885022073650, 4.97797299996805279536919651508, 5.36919440661996387875593296889, 5.45568883557095004276687322825, 5.72044241218869125995973502928, 6.04838804013094700368736240085, 6.24379037902097920300892982671, 6.36220492138492771204754876556, 6.47493504717400522196668687869, 6.88467179314056365041275759683
