| L(s) = 1 | + 4·4-s − 48·11-s + 12·16-s − 80·23-s + 32·25-s + 64·29-s − 128·37-s − 176·43-s − 192·44-s + 96·53-s + 32·64-s + 256·67-s − 32·71-s − 192·79-s − 320·92-s + 128·100-s + 160·107-s − 128·113-s + 256·116-s + 1.02e3·121-s + 127-s + 131-s + 137-s + 139-s − 512·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 4-s − 4.36·11-s + 3/4·16-s − 3.47·23-s + 1.27·25-s + 2.20·29-s − 3.45·37-s − 4.09·43-s − 4.36·44-s + 1.81·53-s + 1/2·64-s + 3.82·67-s − 0.450·71-s − 2.43·79-s − 3.47·92-s + 1.27·100-s + 1.49·107-s − 1.13·113-s + 2.20·116-s + 8.42·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 3.45·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.221071025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221071025\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2:C_4$ | \( 1 - 32 T^{2} + 448 T^{4} - 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 24 T + 354 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 608 T^{2} + 148480 T^{4} - 608 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 832 T^{2} + 326976 T^{4} - 832 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 516 T^{2} + 112038 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 40 T + 1170 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 32 T + 1600 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 3524 T^{2} + 4926598 T^{4} - 3524 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 64 T + 3744 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 5696 T^{2} + 8128 p^{2} T^{4} - 5696 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 88 T + 5506 T^{2} + 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 4100 T^{2} + 5830 p^{2} T^{4} - 4100 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 48 T + 5682 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 4164 T^{2} + 4792038 T^{4} - 4164 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 32 p T^{2} + 28550080 T^{4} - 32 p^{5} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 128 T + 11922 T^{2} - 128 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 6946 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 7872 T^{2} + 69257856 T^{4} - 7872 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 96 T + 13986 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 24708 T^{2} + 247322790 T^{4} - 24708 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 27264 T^{2} + 306701184 T^{4} - 27264 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 11136 T^{2} + 86599488 T^{4} - 11136 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−7.00125942522668782181653555085, −6.85607364898595694231011728511, −6.79397975204649506128641887726, −6.55347110516277650381593716300, −5.98029429157726610797536738886, −5.90303315147217947219512000544, −5.64644558341960268044314668308, −5.39122276620086293579905845160, −5.11038919575293745135258507345, −5.10680879282948097075207173587, −5.00557678304510097978112320939, −4.45626868840280593180938025632, −4.28350013639424029067318197010, −3.78988691764880134229440141651, −3.67053498227267800778549335741, −3.12050547401495266717209023100, −2.95890654069285627084479241507, −2.95206268756440507799760779502, −2.51599179126604180681276097392, −2.11283207130781279273042856999, −1.93478905707711902997454443835, −1.85939385416775124356574256445, −1.19214644824302628491362526929, −0.37232157239706907574116571701, −0.30548638233694494270405231491,
0.30548638233694494270405231491, 0.37232157239706907574116571701, 1.19214644824302628491362526929, 1.85939385416775124356574256445, 1.93478905707711902997454443835, 2.11283207130781279273042856999, 2.51599179126604180681276097392, 2.95206268756440507799760779502, 2.95890654069285627084479241507, 3.12050547401495266717209023100, 3.67053498227267800778549335741, 3.78988691764880134229440141651, 4.28350013639424029067318197010, 4.45626868840280593180938025632, 5.00557678304510097978112320939, 5.10680879282948097075207173587, 5.11038919575293745135258507345, 5.39122276620086293579905845160, 5.64644558341960268044314668308, 5.90303315147217947219512000544, 5.98029429157726610797536738886, 6.55347110516277650381593716300, 6.79397975204649506128641887726, 6.85607364898595694231011728511, 7.00125942522668782181653555085
