L(s) = 1 | − 8·16-s − 4·31-s + 4·43-s + 2·49-s − 4·73-s + 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | − 8·16-s − 4·31-s + 4·43-s + 2·49-s − 4·73-s + 81-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 241^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 241^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09625169364\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09625169364\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
| 241 | \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \) |
good | 2 | \( ( 1 + T^{4} )^{8} \) |
| 5 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 7 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 11 | \( ( 1 + T^{8} )^{4} \) |
| 13 | \( ( 1 + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \) |
| 17 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 19 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 23 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 29 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 31 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 37 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 41 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 43 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 53 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 59 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 61 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 67 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
| 71 | \( 1 - T^{8} + T^{16} - T^{24} + T^{32} \) |
| 73 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 79 | \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \) |
| 83 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \) |
| 89 | \( ( 1 + T^{8} )^{4} \) |
| 97 | \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.95058599179678617891973105398, −2.93185861444184644926174672958, −2.81656968986565822336893575832, −2.79265021463431918563890455634, −2.76734462571796161873970638185, −2.73721776144824284606854038178, −2.65882660046340465268640877301, −2.46536572948297363872433937051, −2.26700261175387134565059522331, −2.22828770461585235388507943151, −2.18529203568086938100292936089, −2.15812642129919159558977290088, −2.05976409916572979869435350999, −1.99465812269000676796825668164, −1.95267202580900828802467861774, −1.93994628129376184274284164000, −1.78369442696156711910489067142, −1.68119734513961924150749710037, −1.58839849175731181443384413306, −1.39667898996689312948828851651, −1.25223685278196553726734775635, −1.02622309788362168630356277211, −1.01271942470610658630927416191, −0.78153479464684803768118330561, −0.44872896704561440138061203752,
0.44872896704561440138061203752, 0.78153479464684803768118330561, 1.01271942470610658630927416191, 1.02622309788362168630356277211, 1.25223685278196553726734775635, 1.39667898996689312948828851651, 1.58839849175731181443384413306, 1.68119734513961924150749710037, 1.78369442696156711910489067142, 1.93994628129376184274284164000, 1.95267202580900828802467861774, 1.99465812269000676796825668164, 2.05976409916572979869435350999, 2.15812642129919159558977290088, 2.18529203568086938100292936089, 2.22828770461585235388507943151, 2.26700261175387134565059522331, 2.46536572948297363872433937051, 2.65882660046340465268640877301, 2.73721776144824284606854038178, 2.76734462571796161873970638185, 2.79265021463431918563890455634, 2.81656968986565822336893575832, 2.93185861444184644926174672958, 2.95058599179678617891973105398
Plot not available for L-functions of degree greater than 10.