| L(s) = 1 | + 2-s − 8·3-s + 3·4-s − 8·6-s + 7·8-s + 20·9-s − 32·11-s − 24·12-s − 9·16-s − 80·17-s + 20·18-s + 56·19-s − 32·22-s − 56·24-s + 92·25-s + 24·27-s − 5·32-s + 256·33-s − 80·34-s + 60·36-s + 56·38-s + 128·41-s − 96·44-s + 72·48-s − 28·49-s + 92·50-s + 640·51-s + ⋯ |
| L(s) = 1 | + 1/2·2-s − 8/3·3-s + 3/4·4-s − 4/3·6-s + 7/8·8-s + 20/9·9-s − 2.90·11-s − 2·12-s − 0.562·16-s − 4.70·17-s + 10/9·18-s + 2.94·19-s − 1.45·22-s − 7/3·24-s + 3.67·25-s + 8/9·27-s − 0.156·32-s + 7.75·33-s − 2.35·34-s + 5/3·36-s + 1.47·38-s + 3.12·41-s − 2.18·44-s + 3/2·48-s − 4/7·49-s + 1.83·50-s + 12.5·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4124000613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4124000613\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T - p T^{2} - p T^{3} + 3 p^{3} T^{4} - p^{3} T^{5} - p^{5} T^{6} - p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 + 4 T + 14 T^{2} + 28 T^{3} + 98 T^{4} + 28 p^{2} T^{5} + 14 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 5 | \( 1 - 92 T^{2} + 5464 T^{4} - 211956 T^{6} + 6231214 T^{8} - 211956 p^{4} T^{10} + 5464 p^{8} T^{12} - 92 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 16 T + 328 T^{2} + 4944 T^{3} + 49230 T^{4} + 4944 p^{2} T^{5} + 328 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 35361044 p^{4} T^{10} + 142936 p^{8} T^{12} - 444 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 40 T + 1308 T^{2} + 31512 T^{3} + 588230 T^{4} + 31512 p^{2} T^{5} + 1308 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 28 T + 90 p T^{2} - 31332 T^{3} + 975266 T^{4} - 31332 p^{2} T^{5} + 90 p^{5} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 412076080 p^{4} T^{10} + 1272156 p^{8} T^{12} - 1744 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 8754768776 p^{4} T^{10} + 6555580 p^{8} T^{12} - 3384 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 12447428312 p^{4} T^{10} + 8438620 p^{8} T^{12} - 3944 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 3512 T^{2} + 9188668 T^{4} - 18622781448 T^{6} + 27544347275206 T^{8} - 18622781448 p^{4} T^{10} + 9188668 p^{8} T^{12} - 3512 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( ( 1 - 64 T + 4956 T^{2} - 221760 T^{3} + 11848326 T^{4} - 221760 p^{2} T^{5} + 4956 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4680 T^{2} - 58016 T^{3} + 10251086 T^{4} - 58016 p^{2} T^{5} + 4680 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 127207295352 p^{4} T^{10} + 39566748 p^{8} T^{12} - 8392 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 18920 T^{2} + 162796828 T^{4} - 840091728600 T^{6} + 2864724835962118 T^{8} - 840091728600 p^{4} T^{10} + 162796828 p^{8} T^{12} - 18920 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( ( 1 - 52 T + 2254 T^{2} + 207508 T^{3} - 19795230 T^{4} + 207508 p^{2} T^{5} + 2254 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 16316 T^{2} + 140172120 T^{4} - 816942037524 T^{6} + 3499102878259502 T^{8} - 816942037524 p^{4} T^{10} + 140172120 p^{8} T^{12} - 16316 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 - 152 T + 22224 T^{2} - 2037320 T^{3} + 158433022 T^{4} - 2037320 p^{2} T^{5} + 22224 p^{4} T^{6} - 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 294531431096 p^{4} T^{10} + 51888284 p^{8} T^{12} - 9864 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 56 T + 18460 T^{2} + 736008 T^{3} + 138223494 T^{4} + 736008 p^{2} T^{5} + 18460 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} - 3106127956152 p^{4} T^{10} + 330869788 p^{8} T^{12} - 24968 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 - 36 T + 16478 T^{2} - 177884 T^{3} + 135298114 T^{4} - 177884 p^{2} T^{5} + 16478 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 256 T + 48252 T^{2} + 6269952 T^{3} + 638304966 T^{4} + 6269952 p^{2} T^{5} + 48252 p^{4} T^{6} + 256 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 1437536 p^{2} T^{5} + 19484 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−7.00060725884540152199769280461, −6.76813598874068242060133006886, −6.70690368386254828997260112102, −6.57362636089948573256769527920, −6.48616986582558272898531218663, −6.04951841765891415464662996156, −5.97136455438983833901146784333, −5.61069342292341079188917965945, −5.58079347019406665365074131610, −5.37318725499127924624946832431, −5.06655906075711888793528675865, −5.00599829828753315537542256157, −4.96527729488269925598314760827, −4.65311898925642541781413155323, −4.44557232353856118308418237111, −4.31043738524039274457625280890, −3.91805967173436858910043170347, −3.55482134082125469948507007129, −2.97495578888134730478562963596, −2.72928896055146672295760503560, −2.60670644644854021553731466612, −2.50383677884877753339977415568, −2.06084243562262991664560995984, −1.01316270294192321577775258844, −0.40113159036813125091222350800,
0.40113159036813125091222350800, 1.01316270294192321577775258844, 2.06084243562262991664560995984, 2.50383677884877753339977415568, 2.60670644644854021553731466612, 2.72928896055146672295760503560, 2.97495578888134730478562963596, 3.55482134082125469948507007129, 3.91805967173436858910043170347, 4.31043738524039274457625280890, 4.44557232353856118308418237111, 4.65311898925642541781413155323, 4.96527729488269925598314760827, 5.00599829828753315537542256157, 5.06655906075711888793528675865, 5.37318725499127924624946832431, 5.58079347019406665365074131610, 5.61069342292341079188917965945, 5.97136455438983833901146784333, 6.04951841765891415464662996156, 6.48616986582558272898531218663, 6.57362636089948573256769527920, 6.70690368386254828997260112102, 6.76813598874068242060133006886, 7.00060725884540152199769280461
Plot not available for L-functions of degree greater than 10.
