| L(s) = 1 | − 8·11-s + 8·37-s − 48·49-s + 16·59-s − 24·61-s + 72·83-s − 64·97-s − 80·107-s − 48·109-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 1.31·37-s − 6.85·49-s + 2.08·59-s − 3.07·61-s + 7.90·83-s − 6.49·97-s − 7.73·107-s − 4.59·109-s + 2.90·121-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 + 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7190409960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7190409960\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{4} - 474 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 82 T^{4} + 28 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 62 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 40 T^{2} + 786 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 292 T^{4} - 25242 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( 1 - 964 T^{4} + 566886 T^{8} - 964 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 40 T^{2} + 594 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 8 T^{2} - 60 T^{3} - 34 T^{4} - 60 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 40 T^{2} + 2034 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 3548 T^{4} + 6433446 T^{8} - 3548 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 - 4868 T^{4} + 16478118 T^{8} - 4868 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 8 T + 32 T^{2} + 232 T^{3} - 6062 T^{4} + 232 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + 10276 T^{4} + 56007654 T^{8} + 10276 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 188 T^{2} + 18726 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 168 T^{2} + 14738 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 36 T + 648 T^{2} - 8604 T^{3} + 89906 T^{4} - 8604 p T^{5} + 648 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 80 T^{2} + 15714 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 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
−3.62423040312052996167900074485, −3.56161775149445892836639474857, −3.55773239329553824790739656088, −3.53741007635323264091487731875, −3.24475419564948829429836604379, −3.03969647143418976612445492624, −2.95715373805915817187388706483, −2.78534996245256578145010585429, −2.77858422860297857283926320515, −2.68410931376114390030519151300, −2.68290672432274998508040849770, −2.41374091945196043742240632275, −2.36929144722319041655457482558, −2.20546580515623896645885719592, −1.84553409624276569199902387916, −1.75715198217115964750620349520, −1.58585953695447658876286856088, −1.51004055869064848731401761575, −1.48206578845124325297493830728, −1.39297681233095946707080251480, −1.00384174072081306226234966140, −0.70077652915045827259313585044, −0.57041734536062553511441666964, −0.24286689726572429557814866291, −0.14455262714612515119651923319,
0.14455262714612515119651923319, 0.24286689726572429557814866291, 0.57041734536062553511441666964, 0.70077652915045827259313585044, 1.00384174072081306226234966140, 1.39297681233095946707080251480, 1.48206578845124325297493830728, 1.51004055869064848731401761575, 1.58585953695447658876286856088, 1.75715198217115964750620349520, 1.84553409624276569199902387916, 2.20546580515623896645885719592, 2.36929144722319041655457482558, 2.41374091945196043742240632275, 2.68290672432274998508040849770, 2.68410931376114390030519151300, 2.77858422860297857283926320515, 2.78534996245256578145010585429, 2.95715373805915817187388706483, 3.03969647143418976612445492624, 3.24475419564948829429836604379, 3.53741007635323264091487731875, 3.55773239329553824790739656088, 3.56161775149445892836639474857, 3.62423040312052996167900074485
Plot not available for L-functions of degree greater than 10.
