| L(s) = 1 | + 2·4-s − 8·7-s + 16-s − 24·19-s − 4·25-s − 16·28-s + 12·31-s − 16·37-s − 16·43-s + 12·49-s − 24·61-s − 2·64-s + 40·67-s + 24·73-s − 48·76-s + 28·79-s − 60·97-s − 8·100-s − 40·109-s − 8·112-s + 52·121-s + 24·124-s + 127-s + 131-s + 192·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s − 3.02·7-s + 1/4·16-s − 5.50·19-s − 4/5·25-s − 3.02·28-s + 2.15·31-s − 2.63·37-s − 2.43·43-s + 12/7·49-s − 3.07·61-s − 1/4·64-s + 4.88·67-s + 2.80·73-s − 5.50·76-s + 3.15·79-s − 6.09·97-s − 4/5·100-s − 3.83·109-s − 0.755·112-s + 4.72·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 16.6·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1696329216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1696329216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 32 T^{2} + 478 T^{4} + 1024 T^{6} - 81341 T^{8} + 1024 p^{2} T^{10} + 478 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2487 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 + 62 T^{2} + 1849 T^{4} + 19406 T^{6} + 39940 T^{8} + 19406 p^{2} T^{10} + 1849 p^{4} T^{12} + 62 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} - 468 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T - 8 T^{2} - 16 T^{3} + 1447 T^{4} - 16 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 20 T^{2} + 1546 T^{4} + 90160 T^{6} - 2184845 T^{8} + 90160 p^{2} T^{10} + 1546 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 8 T - 20 T^{2} - 16 T^{3} + 2455 T^{4} - 16 p T^{5} - 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 152 T^{2} + 13198 T^{4} - 834176 T^{6} + 42212419 T^{8} - 834176 p^{2} T^{10} + 13198 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 158 T^{2} + 13753 T^{4} + 883694 T^{6} + 47672164 T^{8} + 883694 p^{2} T^{10} + 13753 p^{4} T^{12} + 158 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 38 T^{2} - 4727 T^{4} + 30058 T^{6} + 20937316 T^{8} + 30058 p^{2} T^{10} - 4727 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 12 T + 176 T^{2} + 1536 T^{3} + 15591 T^{4} + 1536 p T^{5} + 176 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 176 T^{2} + 15234 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 12 T + 182 T^{2} - 1608 T^{3} + 16131 T^{4} - 1608 p T^{5} + 182 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 14 T + 7 T^{2} - 434 T^{3} + 13996 T^{4} - 434 p T^{5} + 7 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 278 T^{2} + 44473 T^{4} - 5291174 T^{6} + 496693924 T^{8} - 5291174 p^{2} T^{10} + 44473 p^{4} T^{12} - 278 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 30 T + 545 T^{2} + 7350 T^{3} + 79716 T^{4} + 7350 p T^{5} + 545 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.01231145694626596260661584882, −3.91645280064645024538819632154, −3.90934302797270181849912332198, −3.79592634885869907686639689266, −3.68441117320273958694575189038, −3.60684981512190593862992057633, −3.36838057950481315889611143220, −3.29160334072020780800300712337, −3.22809867387315461968946102662, −2.95465489508182444121239513234, −2.72737636194934561065808009161, −2.66541082644325467460553040412, −2.56486668140189334861380391947, −2.50911725625774417764983743211, −2.26105324077548530217071616699, −2.15510393218552044952009364451, −1.98377628951814410332477074584, −1.79434272403426456215033364830, −1.70272564066912518535171548070, −1.49267475292088127285024808914, −1.29068565394466565391515196819, −1.00322691811117479674757191337, −0.49554967855078850422066594096, −0.27073879061932702441261267532, −0.11716644861599742430740308123,
0.11716644861599742430740308123, 0.27073879061932702441261267532, 0.49554967855078850422066594096, 1.00322691811117479674757191337, 1.29068565394466565391515196819, 1.49267475292088127285024808914, 1.70272564066912518535171548070, 1.79434272403426456215033364830, 1.98377628951814410332477074584, 2.15510393218552044952009364451, 2.26105324077548530217071616699, 2.50911725625774417764983743211, 2.56486668140189334861380391947, 2.66541082644325467460553040412, 2.72737636194934561065808009161, 2.95465489508182444121239513234, 3.22809867387315461968946102662, 3.29160334072020780800300712337, 3.36838057950481315889611143220, 3.60684981512190593862992057633, 3.68441117320273958694575189038, 3.79592634885869907686639689266, 3.90934302797270181849912332198, 3.91645280064645024538819632154, 4.01231145694626596260661584882
Plot not available for L-functions of degree greater than 10.
