| L(s) = 1 | + 8·7-s + 2·9-s − 16-s − 12·17-s − 8·25-s − 24·29-s + 8·31-s − 16·37-s − 12·43-s + 8·47-s + 32·49-s + 8·53-s − 16·59-s + 8·61-s + 16·63-s + 8·67-s − 16·73-s − 5·81-s − 16·83-s + 56·89-s + 8·97-s + 16·103-s + 20·107-s − 8·112-s + 28·113-s − 96·119-s + 12·121-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 2/3·9-s − 1/4·16-s − 2.91·17-s − 8/5·25-s − 4.45·29-s + 1.43·31-s − 2.63·37-s − 1.82·43-s + 1.16·47-s + 32/7·49-s + 1.09·53-s − 2.08·59-s + 1.02·61-s + 2.01·63-s + 0.977·67-s − 1.87·73-s − 5/9·81-s − 1.75·83-s + 5.93·89-s + 0.812·97-s + 1.57·103-s + 1.93·107-s − 0.755·112-s + 2.63·113-s − 8.80·119-s + 1.09·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098712054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098712054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 226 T^{4} - 88 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7666 T^{4} + 1040 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 926 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 152 T^{3} - 62 T^{4} - 152 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 456 T^{3} + 6482 T^{4} - 456 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 472 T^{3} + 6898 T^{4} - 472 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 3458 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 80 T^{3} - 4574 T^{4} + 80 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1584 T^{3} + 19346 T^{4} + 1584 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 28 T + 366 T^{2} - 28 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 40 T^{3} - 8414 T^{4} - 40 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.420480113526148456190858538482, −7.77066286457217796851671203475, −7.68985478775935287237194826593, −7.59411723170592654912278126469, −7.20648646356412786371071650270, −7.05364071541796512982191358165, −6.95647628325603408388792763229, −6.35229711959332705653023720912, −6.19074691674378051298019067888, −5.88377863499871455464364409861, −5.47730516189928737245170893962, −5.42836640169361616246243333882, −4.96910290928954594616479780744, −4.64478318045562385908415818609, −4.60588035208488916483924363412, −4.46676605468696305797283336495, −3.84458850088418531275537707719, −3.76758878223743538022469706662, −3.49414023136330298830874490160, −2.82863775180042371616231667831, −2.05893188539416230483846160679, −1.91208173192140746515036717592, −1.82401360944476889042286792134, −1.81780717236730009860813051572, −0.52324226545872310183666597743,
0.52324226545872310183666597743, 1.81780717236730009860813051572, 1.82401360944476889042286792134, 1.91208173192140746515036717592, 2.05893188539416230483846160679, 2.82863775180042371616231667831, 3.49414023136330298830874490160, 3.76758878223743538022469706662, 3.84458850088418531275537707719, 4.46676605468696305797283336495, 4.60588035208488916483924363412, 4.64478318045562385908415818609, 4.96910290928954594616479780744, 5.42836640169361616246243333882, 5.47730516189928737245170893962, 5.88377863499871455464364409861, 6.19074691674378051298019067888, 6.35229711959332705653023720912, 6.95647628325603408388792763229, 7.05364071541796512982191358165, 7.20648646356412786371071650270, 7.59411723170592654912278126469, 7.68985478775935287237194826593, 7.77066286457217796851671203475, 8.420480113526148456190858538482
