| L(s) = 1 | + 4·2-s + 6·4-s − 9-s − 2·11-s − 15·16-s − 4·18-s − 8·22-s − 6·23-s + 9·25-s + 16·29-s − 24·32-s − 6·36-s + 48·37-s − 10·43-s − 12·44-s − 24·46-s + 36·50-s − 32·53-s + 64·58-s − 6·64-s + 2·67-s − 44·71-s + 192·74-s − 34·79-s + 9·81-s − 40·86-s − 36·92-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s − 1/3·9-s − 0.603·11-s − 3.75·16-s − 0.942·18-s − 1.70·22-s − 1.25·23-s + 9/5·25-s + 2.97·29-s − 4.24·32-s − 36-s + 7.89·37-s − 1.52·43-s − 1.80·44-s − 3.53·46-s + 5.09·50-s − 4.39·53-s + 8.40·58-s − 3/4·64-s + 0.244·67-s − 5.22·71-s + 22.3·74-s − 3.82·79-s + 81-s − 4.31·86-s − 3.75·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.65874339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.65874339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 9 T^{2} + 37 T^{4} + 54 T^{6} - 714 T^{8} + 54 p^{2} T^{10} + 37 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + T + 5 T^{2} - 26 T^{3} - 116 T^{4} - 26 p T^{5} + 5 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 8 T^{2} + 10 p T^{4} + 3232 T^{6} - 35021 T^{8} + 3232 p^{2} T^{10} + 10 p^{5} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 29 T^{2} + 552 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 3 T - 13 T^{2} - 72 T^{3} - 252 T^{4} - 72 p T^{5} - 13 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 80 T^{2} + 3298 T^{4} - 94400 T^{6} + 2548483 T^{8} - 94400 p^{2} T^{10} + 3298 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 41 | \( 1 - 125 T^{2} + 8593 T^{4} - 458750 T^{6} + 20357638 T^{8} - 458750 p^{2} T^{10} + 8593 p^{4} T^{12} - 125 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 5 T - 41 T^{2} - 100 T^{3} + 1432 T^{4} - 100 p T^{5} - 41 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 18 T^{2} - 1885 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 59 | \( 1 + 135 T^{2} + 7993 T^{4} + 441450 T^{6} + 29135238 T^{8} + 441450 p^{2} T^{10} + 7993 p^{4} T^{12} + 135 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 173 T^{2} + 15241 T^{4} - 1253558 T^{6} + 90525694 T^{8} - 1253558 p^{2} T^{10} + 15241 p^{4} T^{12} - 173 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - T - 107 T^{2} + 26 T^{3} + 7108 T^{4} + 26 p T^{5} - 107 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 11 T + 146 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 113 T^{2} + 12564 T^{4} + 113 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 17 T + 85 T^{2} + 782 T^{3} + 12544 T^{4} + 782 p T^{5} + 85 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 106 T^{2} + 4347 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 72 T^{2} + 13358 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 113 T^{2} + 7165 T^{4} + 1493182 T^{6} - 177267986 T^{8} + 1493182 p^{2} T^{10} + 7165 p^{4} T^{12} - 113 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.38303928534351719761247096073, −4.34053591584514458851164672089, −4.23835013706424996954253663111, −4.21597783619640694108830673086, −4.20521846777214217265506078981, −3.69205632844268688429982993152, −3.40042407276845395119462250296, −3.34176831577477017345753538623, −3.27267860253371788755887884359, −3.23934125705889452362582107078, −2.96725108892900320806928862756, −2.91298019207988420572649471610, −2.81651781903302358397688872471, −2.81280613820803625826916192113, −2.72901771893979196457397732080, −2.27054593071924833723203692791, −2.13130240694685371382282877735, −2.01642643235596971097763209076, −1.79729857462105952002621009340, −1.69762150722892890105460943921, −1.17668171270291697187632242421, −1.03593458057721478185447624997, −0.916552179839288339412655991224, −0.55114083504725354834598234973, −0.36504194591740014477172548498,
0.36504194591740014477172548498, 0.55114083504725354834598234973, 0.916552179839288339412655991224, 1.03593458057721478185447624997, 1.17668171270291697187632242421, 1.69762150722892890105460943921, 1.79729857462105952002621009340, 2.01642643235596971097763209076, 2.13130240694685371382282877735, 2.27054593071924833723203692791, 2.72901771893979196457397732080, 2.81280613820803625826916192113, 2.81651781903302358397688872471, 2.91298019207988420572649471610, 2.96725108892900320806928862756, 3.23934125705889452362582107078, 3.27267860253371788755887884359, 3.34176831577477017345753538623, 3.40042407276845395119462250296, 3.69205632844268688429982993152, 4.20521846777214217265506078981, 4.21597783619640694108830673086, 4.23835013706424996954253663111, 4.34053591584514458851164672089, 4.38303928534351719761247096073
Plot not available for L-functions of degree greater than 10.
