| L(s) = 1 | − 4·13-s − 48·47-s − 8·67-s + 25·81-s + 64·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 74·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 7.00·47-s − 0.977·67-s + 25/9·81-s + 6.30·103-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 − 5.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09028174610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09028174610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 25 T^{4} + 313 T^{8} - 25 p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( 1 + 7 T^{4} + 81 T^{8} + 7 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( 1 + 79 T^{4} + 3081 T^{8} + 79 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 241 T^{4} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 53 T^{2} + 1293 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 425 T^{4} - 110007 T^{8} - 425 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( 1 - 977 T^{4} + 1595313 T^{8} - 977 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 553 T^{4} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 100 T^{4} + 2842278 T^{8} + 100 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 161 T^{2} + 11673 T^{4} - 161 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( 1 + 1918 T^{4} + 28577763 T^{8} + 1918 p^{4} T^{12} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 1154 T^{4} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 10850 T^{4} + 59507043 T^{8} - 10850 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 11234 T^{4} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 281 T^{2} + 33093 T^{4} - 281 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 29561 T^{4} + 393470961 T^{8} - 29561 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.18161737594669107573322593312, −4.10110637650114608867908591285, −3.89683804355364559199979048448, −3.75058830649119998558504152530, −3.55769247597770214229581275850, −3.50683857503536937842962423548, −3.31705089073633393887951936279, −3.29479894814413468868228078950, −3.19610419787389980455006696047, −3.05577689334914296891059467017, −3.00620030240724282320247987777, −2.57222349132963907635741016582, −2.51364568826026558120887510340, −2.33084344341206822431830156601, −2.19170843161354208337698588313, −2.17685908132197555315350438866, −2.03522740070892476073919766778, −1.68445607394667279610990932752, −1.63063467078515983551882176778, −1.41363716873928686179859979738, −1.11342949262285980345995688459, −1.07783878324355221882519769283, −0.883434205022067840650809152732, −0.23174532763892487411785869860, −0.06624288141569739622302419792,
0.06624288141569739622302419792, 0.23174532763892487411785869860, 0.883434205022067840650809152732, 1.07783878324355221882519769283, 1.11342949262285980345995688459, 1.41363716873928686179859979738, 1.63063467078515983551882176778, 1.68445607394667279610990932752, 2.03522740070892476073919766778, 2.17685908132197555315350438866, 2.19170843161354208337698588313, 2.33084344341206822431830156601, 2.51364568826026558120887510340, 2.57222349132963907635741016582, 3.00620030240724282320247987777, 3.05577689334914296891059467017, 3.19610419787389980455006696047, 3.29479894814413468868228078950, 3.31705089073633393887951936279, 3.50683857503536937842962423548, 3.55769247597770214229581275850, 3.75058830649119998558504152530, 3.89683804355364559199979048448, 4.10110637650114608867908591285, 4.18161737594669107573322593312
Plot not available for L-functions of degree greater than 10.
