| L(s) = 1 | + 6·9-s + 6·11-s + 2·19-s + 4·29-s + 2·31-s − 60·41-s − 5·49-s − 2·59-s + 24·61-s + 24·71-s − 14·79-s + 27·81-s + 28·89-s + 36·99-s + 24·101-s + 16·109-s + 29·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + ⋯ |
| L(s) = 1 | + 2·9-s + 1.80·11-s + 0.458·19-s + 0.742·29-s + 0.359·31-s − 9.37·41-s − 5/7·49-s − 0.260·59-s + 3.07·61-s + 2.84·71-s − 1.57·79-s + 3·81-s + 2.96·89-s + 3.61·99-s + 2.38·101-s + 1.53·109-s + 2.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.23·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.806464363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.806464363\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 45 T^{2} + 1069 T^{4} - 17010 T^{6} + 245190 T^{8} - 17010 p^{2} T^{10} + 1069 p^{4} T^{12} - 45 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - T - 23 T^{2} + 14 T^{3} + 196 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2}( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 29 | \( ( 1 - T + 44 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - T - 47 T^{2} + 14 T^{3} + 1312 T^{4} + 14 p T^{5} - 47 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 17 T^{2} - 1367 T^{4} + 18394 T^{6} + 472534 T^{8} + 18394 p^{2} T^{10} - 1367 p^{4} T^{12} - 17 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 125 T^{2} + 7248 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 3 p T^{2} + 10849 T^{4} - 13842 p T^{6} + 32780214 T^{8} - 13842 p^{3} T^{10} + 10849 p^{4} T^{12} - 3 p^{7} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 125 T^{2} + 6229 T^{4} - 472250 T^{6} + 36503710 T^{8} - 472250 p^{2} T^{10} + 6229 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + T - 103 T^{2} - 14 T^{3} + 7276 T^{4} - 14 p T^{5} - 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 43 T^{2} + 252 T^{3} - 2304 T^{4} + 252 p T^{5} + 43 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 62 T^{2} + 2113 T^{4} + 449314 T^{6} - 34410476 T^{8} + 449314 p^{2} T^{10} + 2113 p^{4} T^{12} - 62 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 245 T^{2} + 34717 T^{4} - 3589250 T^{6} + 292223398 T^{8} - 3589250 p^{2} T^{10} + 34717 p^{4} T^{12} - 245 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 7 T - 107 T^{2} - 14 T^{3} + 14224 T^{4} - 14 p T^{5} - 107 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 245 T^{2} + 28656 T^{4} + 245 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.59633971808217535230569560533, −4.51545782885505770942194586838, −4.23256867405151727780494334325, −4.11130623690261620481840133235, −3.96224881433107441035677121778, −3.71428345267759365227908469390, −3.65244688997925271283800864836, −3.62245920364958077236720398207, −3.35777809419717619305855875426, −3.33696928833344014879654544925, −3.27308483511750441742674265190, −3.16752585559258338742474332763, −3.06303144955831308829779934307, −2.52487229810477797383060906819, −2.32944063059052958482943840356, −2.07469347460716453669495629770, −2.04912056122037259148078230545, −1.99640258035874522604597726268, −1.96944228998277801746996768337, −1.44736867702646672723102012219, −1.37418934929991098085460143080, −1.18091705287529041769252528555, −1.06355585062648644404595131153, −0.69274806808699289534775993632, −0.24770195952458898628909719282,
0.24770195952458898628909719282, 0.69274806808699289534775993632, 1.06355585062648644404595131153, 1.18091705287529041769252528555, 1.37418934929991098085460143080, 1.44736867702646672723102012219, 1.96944228998277801746996768337, 1.99640258035874522604597726268, 2.04912056122037259148078230545, 2.07469347460716453669495629770, 2.32944063059052958482943840356, 2.52487229810477797383060906819, 3.06303144955831308829779934307, 3.16752585559258338742474332763, 3.27308483511750441742674265190, 3.33696928833344014879654544925, 3.35777809419717619305855875426, 3.62245920364958077236720398207, 3.65244688997925271283800864836, 3.71428345267759365227908469390, 3.96224881433107441035677121778, 4.11130623690261620481840133235, 4.23256867405151727780494334325, 4.51545782885505770942194586838, 4.59633971808217535230569560533
Plot not available for L-functions of degree greater than 10.
