| L(s) = 1 | − 10·5-s + 46·9-s + 20·11-s + 19·25-s + 4·31-s − 460·45-s − 360·49-s − 200·55-s + 60·59-s − 564·71-s + 1.05e3·81-s + 68·89-s + 920·99-s − 8·121-s + 90·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 40·155-s + 157-s + 163-s + 167-s − 1.07e3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·5-s + 46/9·9-s + 1.81·11-s + 0.759·25-s + 4/31·31-s − 10.2·45-s − 7.34·49-s − 3.63·55-s + 1.01·59-s − 7.94·71-s + 13.0·81-s + 0.764·89-s + 9.29·99-s − 0.0661·121-s + 0.719·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.258·155-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 6.36·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.618841370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.618841370\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + p T + 28 T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | \( ( 1 - 10 T + 14 p T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 23 T^{2} + 266 T^{4} - 23 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 90 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 538 T^{2} + 126658 T^{4} + 538 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 138 T^{2} + 170786 T^{4} + 138 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 312 T^{2} + 2478 T^{4} - 312 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1607 T^{2} + 1142890 T^{4} - 1607 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1908 T^{2} + 1970310 T^{4} - 1908 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - T + 538 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2433 T^{2} + 3746736 T^{4} - 2433 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 2076 T^{2} + 6640374 T^{4} - 2076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 2652 T^{2} + 5845910 T^{4} + 2652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4218 T^{2} + 14188146 T^{4} - 4218 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 9284 T^{2} + 36605926 T^{4} - 9284 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 15 T + 5634 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 13220 T^{2} + 70920934 T^{4} - 13220 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 9543 T^{2} + 45661098 T^{4} - 9543 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 141 T + 14798 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 15754 T^{2} + 118705186 T^{4} + 15754 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 7248 T^{2} + 58326366 T^{4} + 7248 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 16092 T^{2} + 129567830 T^{4} + 16092 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 17 T + 14530 T^{2} - 17 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 30545 T^{2} + 405157024 T^{4} - 30545 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.09333936237769851727911482092, −4.08991265768666513191588469571, −3.97459198997672009813444270896, −3.91442027954074337249336098082, −3.58515201815719175266220105493, −3.53545012925948541761158399128, −3.46851076809449796551295277570, −3.33748744852453444789723044858, −3.08367802996651315288453341477, −2.88051578025589975932147821469, −2.81884225854934074989630320348, −2.68877623835571956553190164830, −2.60730531549130592294615803246, −1.93649944260838595075886511595, −1.84707423226630560173386717166, −1.81254730700973635395168306749, −1.70093552601923862872142258231, −1.59881532105997986132305135627, −1.56832798518926078813497565937, −1.13699804772074606412569274001, −1.12437914452120170645594115221, −1.08305604845238391986077280532, −0.60198046082485043388583128475, −0.27264751865328477857404830933, −0.16331539768783554335127380069,
0.16331539768783554335127380069, 0.27264751865328477857404830933, 0.60198046082485043388583128475, 1.08305604845238391986077280532, 1.12437914452120170645594115221, 1.13699804772074606412569274001, 1.56832798518926078813497565937, 1.59881532105997986132305135627, 1.70093552601923862872142258231, 1.81254730700973635395168306749, 1.84707423226630560173386717166, 1.93649944260838595075886511595, 2.60730531549130592294615803246, 2.68877623835571956553190164830, 2.81884225854934074989630320348, 2.88051578025589975932147821469, 3.08367802996651315288453341477, 3.33748744852453444789723044858, 3.46851076809449796551295277570, 3.53545012925948541761158399128, 3.58515201815719175266220105493, 3.91442027954074337249336098082, 3.97459198997672009813444270896, 4.08991265768666513191588469571, 4.09333936237769851727911482092
Plot not available for L-functions of degree greater than 10.
