| L(s) = 1 | − 8·4-s + 8·9-s − 16·11-s + 36·16-s − 32·23-s + 40·25-s − 64·36-s + 32·37-s + 128·44-s + 56·53-s − 120·64-s − 24·67-s + 8·71-s + 12·81-s + 256·92-s − 128·99-s − 320·100-s − 48·113-s + 152·121-s + 127-s + 131-s + 137-s + 139-s + 288·144-s − 256·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4·4-s + 8/3·9-s − 4.82·11-s + 9·16-s − 6.67·23-s + 8·25-s − 10.6·36-s + 5.26·37-s + 19.2·44-s + 7.69·53-s − 15·64-s − 2.93·67-s + 0.949·71-s + 4/3·81-s + 26.6·92-s − 12.8·99-s − 32·100-s − 4.51·113-s + 13.8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 24·144-s − 21.0·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07199745161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07199745161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 7 | \( 1 \) |
| 11 | \( ( 1 + 8 T + 20 T^{2} - 40 T^{3} - 362 T^{4} - 40 p T^{5} + 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| good | 3 | \( ( 1 - 4 T + 2 p T^{2} + 2 T^{3} - 17 T^{4} + 2 p T^{5} + 2 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2}( 1 + 4 T + 2 p T^{2} - 2 T^{3} - 17 T^{4} - 2 p T^{5} + 2 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 4 p T^{2} + 208 T^{4} - 1472 T^{6} + 1634 p T^{8} - 1472 p^{2} T^{10} + 208 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 60 T^{2} + 1810 T^{4} + 37104 T^{6} + 559467 T^{8} + 37104 p^{2} T^{10} + 1810 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 44 T^{2} + 1060 T^{4} + 13076 T^{6} + 175606 T^{8} + 13076 p^{2} T^{10} + 1060 p^{4} T^{12} + 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 72 T^{2} + 2260 T^{4} + 47412 T^{6} + 896994 T^{8} + 47412 p^{2} T^{10} + 2260 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 8 T + 86 T^{2} + 518 T^{3} + 2884 T^{4} + 518 p T^{5} + 86 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 196 T^{2} + 17650 T^{4} - 953200 T^{6} + 33744235 T^{8} - 953200 p^{2} T^{10} + 17650 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 84 T^{2} + 3604 T^{4} - 156396 T^{6} + 5930646 T^{8} - 156396 p^{2} T^{10} + 3604 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 8 T + 160 T^{2} - 890 T^{3} + 9100 T^{4} - 890 p T^{5} + 160 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 128 T^{2} + 10456 T^{4} + 593396 T^{6} + 27172090 T^{8} + 593396 p^{2} T^{10} + 10456 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 100 T^{2} + 8356 T^{4} - 517756 T^{6} + 23891542 T^{8} - 517756 p^{2} T^{10} + 8356 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 296 T^{2} + 41212 T^{4} - 3509912 T^{6} + 199565062 T^{8} - 3509912 p^{2} T^{10} + 41212 p^{4} T^{12} - 296 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 14 T + 236 T^{2} - 1904 T^{3} + 18316 T^{4} - 1904 p T^{5} + 236 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - 176 T^{2} + 12658 T^{4} - 310940 T^{6} - 1019165 T^{8} - 310940 p^{2} T^{10} + 12658 p^{4} T^{12} - 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 188 T^{2} + 21226 T^{4} + 1882352 T^{6} + 130642675 T^{8} + 1882352 p^{2} T^{10} + 21226 p^{4} T^{12} + 188 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 + 6 T + 214 T^{2} + 1158 T^{3} + 20115 T^{4} + 1158 p T^{5} + 214 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 2 T + 110 T^{2} - 764 T^{3} + 8800 T^{4} - 764 p T^{5} + 110 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 380 T^{2} + 71728 T^{4} + 8798696 T^{6} + 759114346 T^{8} + 8798696 p^{2} T^{10} + 71728 p^{4} T^{12} + 380 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 232 T^{2} + 31330 T^{4} - 2644588 T^{6} + 211607707 T^{8} - 2644588 p^{2} T^{10} + 31330 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 + 404 T^{2} + 73300 T^{4} + 8358284 T^{6} + 744742582 T^{8} + 8358284 p^{2} T^{10} + 73300 p^{4} T^{12} + 404 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 - 256 T^{2} + 40372 T^{4} - 4329424 T^{6} + 416141830 T^{8} - 4329424 p^{2} T^{10} + 40372 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 132 T^{2} + 34546 T^{4} - 2728224 T^{6} + 439608027 T^{8} - 2728224 p^{2} T^{10} + 34546 p^{4} T^{12} - 132 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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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
−2.50653422115915742205488504242, −2.46404760522401013197897898583, −2.44949389915050966642520724315, −2.43858004366927341766893102381, −2.38979279419576225865530518234, −2.25482807113572142830130905864, −2.17162685400863230394580461587, −2.13439456372299629726527988560, −1.98487973423324115163027718469, −1.75234228523780288016581684567, −1.56976949074715005063296826491, −1.56717421279204087130398551983, −1.44457951282400865636534871332, −1.40048319495272688002495201231, −1.30767683655954480293294324587, −1.11350030530876399038398925022, −1.09275074162423425091301645533, −1.00357638863020341865507619821, −0.923038036608626840282288375545, −0.853531050041188999740551319944, −0.72739394603641398179615028384, −0.43209907779437602848677662617, −0.42182088750560705204079943112, −0.22106118038638732114123523929, −0.03999687251798425386462523067,
0.03999687251798425386462523067, 0.22106118038638732114123523929, 0.42182088750560705204079943112, 0.43209907779437602848677662617, 0.72739394603641398179615028384, 0.853531050041188999740551319944, 0.923038036608626840282288375545, 1.00357638863020341865507619821, 1.09275074162423425091301645533, 1.11350030530876399038398925022, 1.30767683655954480293294324587, 1.40048319495272688002495201231, 1.44457951282400865636534871332, 1.56717421279204087130398551983, 1.56976949074715005063296826491, 1.75234228523780288016581684567, 1.98487973423324115163027718469, 2.13439456372299629726527988560, 2.17162685400863230394580461587, 2.25482807113572142830130905864, 2.38979279419576225865530518234, 2.43858004366927341766893102381, 2.44949389915050966642520724315, 2.46404760522401013197897898583, 2.50653422115915742205488504242
Plot not available for L-functions of degree greater than 10.
