| L(s) = 1 | + 8·7-s + 64·13-s − 112·19-s − 64·25-s + 32·31-s − 152·37-s − 88·43-s + 4·49-s + 52·61-s − 40·67-s + 448·73-s + 104·79-s + 512·91-s − 32·97-s − 112·103-s − 128·109-s − 52·121-s + 127-s + 131-s − 896·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 4.92·13-s − 5.89·19-s − 2.55·25-s + 1.03·31-s − 4.10·37-s − 2.04·43-s + 4/49·49-s + 0.852·61-s − 0.597·67-s + 6.13·73-s + 1.31·79-s + 5.62·91-s − 0.329·97-s − 1.08·103-s − 1.17·109-s − 0.429·121-s + 0.00787·127-s + 0.00763·131-s − 6.73·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5232199661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5232199661\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 64 T^{2} + 413 p T^{4} + 49984 T^{6} + 1174336 T^{8} + 49984 p^{4} T^{10} + 413 p^{9} T^{12} + 64 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 4 T + 22 T^{2} + 416 T^{3} - 3149 T^{4} + 416 p^{2} T^{5} + 22 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 26 T^{2} - 13965 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T + 457 T^{2} - 7328 T^{3} + 119872 T^{4} - 7328 p^{2} T^{5} + 457 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 256 T^{2} + 31551 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 28 T + 810 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 842 T^{2} + 429123 T^{4} + 842 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 1312 T^{2} - 68879 T^{4} + 492867232 T^{6} + 1487178071104 T^{8} + 492867232 p^{4} T^{10} - 68879 p^{8} T^{12} + 1312 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 8 T - 897 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 38 T + 1371 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 + 4960 T^{2} + 12861886 T^{4} + 30197432320 T^{6} + 60317717755075 T^{8} + 30197432320 p^{4} T^{10} + 12861886 p^{8} T^{12} + 4960 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 44 T - 1274 T^{2} - 21472 T^{3} + 3305635 T^{4} - 21472 p^{2} T^{5} - 1274 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4130 T^{2} + 12177219 T^{4} + 4130 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3424 T^{2} + 8198754 T^{4} - 3424 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 1828 T^{2} - 16689686 T^{4} - 7683910256 T^{6} + 213930160568755 T^{8} - 7683910256 p^{4} T^{10} - 16689686 p^{8} T^{12} + 1828 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 13 T - 3552 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 20 T - 7706 T^{2} - 17440 T^{3} + 43760515 T^{4} - 17440 p^{2} T^{5} - 7706 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 9652 T^{2} + 49230438 T^{4} - 9652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 112 T + 13551 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 52 T - 5162 T^{2} + 240032 T^{3} + 6048211 T^{4} + 240032 p^{2} T^{5} - 5162 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 16612 T^{2} + 139485994 T^{4} + 690326743696 T^{6} + 3275935507518835 T^{8} + 690326743696 p^{4} T^{10} + 139485994 p^{8} T^{12} + 16612 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 7456 T^{2} + 104845119 T^{4} - 7456 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T - 11714 T^{2} - 109568 T^{3} + 52342915 T^{4} - 109568 p^{2} T^{5} - 11714 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.85934055445010938370489542871, −3.78861221496925421036737915704, −3.71294412411388168812906000977, −3.68161853251145563280570648252, −3.67250789649088984529354782896, −3.25240198877513476623852414390, −3.10605074120820781261867836708, −3.07354869556069022659905132477, −2.84902591846775453510515097215, −2.57370222707784948711239724145, −2.51856184171464418739147685321, −2.25669068901246197947585549699, −2.13357193701476544533653830569, −2.10377606545477995893260381974, −1.78351862628379756661445617041, −1.76919610286636593790306184069, −1.76440378832706522845086197840, −1.57608946114708295516263070857, −1.42237559260859172327735412293, −1.19547331783757989666477646572, −0.911952933556152353326940429159, −0.833741795283597325032273848170, −0.53877430931204847784875041503, −0.18688006939532403264718423005, −0.083029602682387342443116617774,
0.083029602682387342443116617774, 0.18688006939532403264718423005, 0.53877430931204847784875041503, 0.833741795283597325032273848170, 0.911952933556152353326940429159, 1.19547331783757989666477646572, 1.42237559260859172327735412293, 1.57608946114708295516263070857, 1.76440378832706522845086197840, 1.76919610286636593790306184069, 1.78351862628379756661445617041, 2.10377606545477995893260381974, 2.13357193701476544533653830569, 2.25669068901246197947585549699, 2.51856184171464418739147685321, 2.57370222707784948711239724145, 2.84902591846775453510515097215, 3.07354869556069022659905132477, 3.10605074120820781261867836708, 3.25240198877513476623852414390, 3.67250789649088984529354782896, 3.68161853251145563280570648252, 3.71294412411388168812906000977, 3.78861221496925421036737915704, 3.85934055445010938370489542871
Plot not available for L-functions of degree greater than 10.
