| L(s) = 1 | + 30·9-s + 128·11-s − 112·13-s + 320·23-s + 196·25-s − 496·37-s + 492·49-s − 112·61-s + 320·71-s + 616·73-s + 171·81-s − 2.68e3·83-s + 2.85e3·97-s + 3.84e3·99-s + 6.14e3·107-s + 5.26e3·109-s − 3.36e3·117-s + 5.08e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.43e4·143-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 10/9·9-s + 3.50·11-s − 2.38·13-s + 2.90·23-s + 1.56·25-s − 2.20·37-s + 1.43·49-s − 0.235·61-s + 0.534·71-s + 0.987·73-s + 0.234·81-s − 3.55·83-s + 2.98·97-s + 3.89·99-s + 5.55·107-s + 4.62·109-s − 2.65·117-s + 3.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 8.38·143-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(13.76529564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.76529564\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 10 p T^{2} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 774 p^{2} T^{4} - 196 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 492 T^{2} + 102278 T^{4} - 492 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 64 T + 3602 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 56 T + 3834 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6916 T^{2} + 54727878 T^{4} - 6916 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 3588 T^{2} + 29873654 T^{4} + 3588 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 160 T + 29390 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 28900 T^{2} + 841043958 T^{4} - 28900 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112780 T^{2} + 4945356198 T^{4} - 112780 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 248 T + 50826 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 198884 T^{2} + 18596196390 T^{4} - 198884 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 174108 T^{2} + 16916734358 T^{4} - 174108 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 121630 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 133508 T^{2} - 4558541226 T^{4} - 133508 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 408658 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 56 T + 66330 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 690876 T^{2} + 254657072438 T^{4} - 690876 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 160 T - 117778 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 308 T + 608214 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 257100 T^{2} + 501931869158 T^{4} - 257100 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1344 T + 1480162 T^{2} + 1344 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1957732 T^{2} + 1892598888294 T^{4} - 1957732 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1428 T + 2249126 T^{2} - 1428 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.08661821847927365707109563350, −7.02810638280224190806111958111, −6.52040222172172483227674271642, −6.50896438758146112759215243498, −6.12138820555734170518341523687, −5.99262089468109186013877575751, −5.45452711577922196479755727110, −5.31001031852792130673600937803, −4.96421591869219267012313800353, −4.76281912752652846075404484698, −4.67597953801396356894855009897, −4.39147970064097286580958785675, −4.21390535504336320921433460897, −3.58935434557868723340222029755, −3.58432429235747296709515203147, −3.47803236791332249941671566205, −2.90664326841989751315669595374, −2.75336475172500638270112066719, −2.37571651597896855259152482260, −1.82040045636014187829203214168, −1.58227249026353971826223754127, −1.57247457312103759039595911778, −0.855264708230510137018647793822, −0.69105703143152263953657608595, −0.56412099025881594472420908146,
0.56412099025881594472420908146, 0.69105703143152263953657608595, 0.855264708230510137018647793822, 1.57247457312103759039595911778, 1.58227249026353971826223754127, 1.82040045636014187829203214168, 2.37571651597896855259152482260, 2.75336475172500638270112066719, 2.90664326841989751315669595374, 3.47803236791332249941671566205, 3.58432429235747296709515203147, 3.58935434557868723340222029755, 4.21390535504336320921433460897, 4.39147970064097286580958785675, 4.67597953801396356894855009897, 4.76281912752652846075404484698, 4.96421591869219267012313800353, 5.31001031852792130673600937803, 5.45452711577922196479755727110, 5.99262089468109186013877575751, 6.12138820555734170518341523687, 6.50896438758146112759215243498, 6.52040222172172483227674271642, 7.02810638280224190806111958111, 7.08661821847927365707109563350
