| L(s) = 1 | + 4·4-s + 12·7-s + 4·13-s − 20·16-s + 44·25-s + 48·28-s − 96·31-s − 56·37-s − 240·43-s + 2·49-s + 16·52-s − 20·61-s − 160·64-s + 228·67-s + 232·73-s + 660·79-s + 48·91-s + 124·97-s + 176·100-s − 348·103-s − 560·109-s − 240·112-s − 124·121-s − 384·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 4-s + 12/7·7-s + 4/13·13-s − 5/4·16-s + 1.75·25-s + 12/7·28-s − 3.09·31-s − 1.51·37-s − 5.58·43-s + 2/49·49-s + 4/13·52-s − 0.327·61-s − 5/2·64-s + 3.40·67-s + 3.17·73-s + 8.35·79-s + 0.527·91-s + 1.27·97-s + 1.75·100-s − 3.37·103-s − 5.13·109-s − 2.14·112-s − 1.02·121-s − 3.09·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6560946588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6560946588\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{4} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 44 T^{2} + 922 T^{4} + 10384 T^{6} - 572429 T^{8} + 10384 p^{4} T^{10} + 922 p^{8} T^{12} - 44 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 6 T + 53 T^{2} - 246 T^{3} - 132 T^{4} - 246 p^{2} T^{5} + 53 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 124 T^{2} + 250 T^{4} - 1755344 T^{6} - 210683021 T^{8} - 1755344 p^{4} T^{10} + 250 p^{8} T^{12} + 124 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 2 T - 155 T^{2} + 358 T^{3} - 3956 T^{4} + 358 p^{2} T^{5} - 155 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 140 T^{2} + 136662 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 695 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 + 604 T^{2} - 268070 T^{4} + 44215216 T^{6} + 220143229459 T^{8} + 44215216 p^{4} T^{10} - 268070 p^{8} T^{12} + 604 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 2468 T^{2} + 3338026 T^{4} - 3303260048 T^{6} + 2770686029299 T^{8} - 3303260048 p^{4} T^{10} + 3338026 p^{8} T^{12} - 2468 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 48 T + 2 p^{2} T^{2} + 55392 T^{3} + 1146243 T^{4} + 55392 p^{2} T^{5} + 2 p^{6} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 14 T + 2607 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 - 3140 T^{2} + 4692298 T^{4} + 1520450800 T^{6} - 9144788425517 T^{8} + 1520450800 p^{4} T^{10} + 4692298 p^{8} T^{12} - 3140 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 120 T + 9458 T^{2} + 558960 T^{3} + 27153363 T^{4} + 558960 p^{2} T^{5} + 9458 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 + 4444 T^{2} + 113030 p T^{4} + 20786205616 T^{6} + 88238287231699 T^{8} + 20786205616 p^{4} T^{10} + 113030 p^{9} T^{12} + 4444 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 508 T^{2} + 14912358 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 6076 T^{2} + 12975610 T^{4} - 1777570256 T^{6} - 20949143941901 T^{8} - 1777570256 p^{4} T^{10} + 12975610 p^{8} T^{12} + 6076 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 10 T - 7187 T^{2} - 1550 T^{3} + 38882428 T^{4} - 1550 p^{2} T^{5} - 7187 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 114 T + 10553 T^{2} - 709194 T^{3} + 37996068 T^{4} - 709194 p^{2} T^{5} + 10553 p^{4} T^{6} - 114 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 292 T^{2} + 42446598 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 58 T + 10779 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 330 T + 54917 T^{2} - 6143610 T^{3} + 534190908 T^{4} - 6143610 p^{2} T^{5} + 54917 p^{4} T^{6} - 330 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 26116 T^{2} + 416905450 T^{4} + 4445553374224 T^{6} + 35678198362101619 T^{8} + 4445553374224 p^{4} T^{10} + 416905450 p^{8} T^{12} + 26116 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 30668 T^{2} + 360580758 T^{4} + 30668 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 62 T - 4415 T^{2} + 654658 T^{3} - 56486396 T^{4} + 654658 p^{2} T^{5} - 4415 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−5.04373790036654060971349926472, −5.00476918663550773086925554435, −4.72680025937851731109375845308, −4.61875402377880230123497113787, −4.33507828092989860693846147173, −4.19011241054358338100753107429, −3.86055112355454570571666952679, −3.78590324604135125911009306341, −3.65846171821206918164953761012, −3.57351000632698981319495311175, −3.42539530671263134984636664963, −3.25834627777227677426081198121, −3.02093906181800227278008376668, −2.75573378064814627840140636132, −2.62138137609155731920359415889, −2.40182115931470279194052767160, −1.97776269187582703930780677367, −1.89290954619906552714000529828, −1.87289409834048051288783672264, −1.86168510156065146166439729596, −1.63371409745384988884036152580, −1.17588086147620793719981517604, −0.808448795261808097621852291926, −0.65388552550721329184128716579, −0.07516104226426905369482547064,
0.07516104226426905369482547064, 0.65388552550721329184128716579, 0.808448795261808097621852291926, 1.17588086147620793719981517604, 1.63371409745384988884036152580, 1.86168510156065146166439729596, 1.87289409834048051288783672264, 1.89290954619906552714000529828, 1.97776269187582703930780677367, 2.40182115931470279194052767160, 2.62138137609155731920359415889, 2.75573378064814627840140636132, 3.02093906181800227278008376668, 3.25834627777227677426081198121, 3.42539530671263134984636664963, 3.57351000632698981319495311175, 3.65846171821206918164953761012, 3.78590324604135125911009306341, 3.86055112355454570571666952679, 4.19011241054358338100753107429, 4.33507828092989860693846147173, 4.61875402377880230123497113787, 4.72680025937851731109375845308, 5.00476918663550773086925554435, 5.04373790036654060971349926472
Plot not available for L-functions of degree greater than 10.
