| L(s) = 1 | − 4·2-s + 4·3-s + 6·4-s − 16·6-s − 4·8-s + 6·9-s + 24·12-s + 3·16-s + 16·17-s − 24·18-s − 16·24-s − 2·25-s − 4·27-s − 8·31-s − 64·34-s + 36·36-s + 24·37-s + 8·41-s + 12·48-s − 2·49-s + 8·50-s + 64·51-s + 16·54-s + 32·62-s − 28·64-s + 8·67-s + 96·68-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 3·4-s − 6.53·6-s − 1.41·8-s + 2·9-s + 6.92·12-s + 3/4·16-s + 3.88·17-s − 5.65·18-s − 3.26·24-s − 2/5·25-s − 0.769·27-s − 1.43·31-s − 10.9·34-s + 6·36-s + 3.94·37-s + 1.24·41-s + 1.73·48-s − 2/7·49-s + 1.13·50-s + 8.96·51-s + 2.17·54-s + 4.06·62-s − 7/2·64-s + 0.977·67-s + 11.6·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9224744058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9224744058\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| good | 2 | $D_{4}$ | \( ( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - 68 T^{2} + 2086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2854 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 14326 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4854 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 7894 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 10950 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 250 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.41658315979107512340847711271, −7.35065036165764919889751796892, −6.44812847748733345634131191308, −6.38360144564905115530464059347, −6.33146585074858009467525168055, −6.14218864974407392950114274744, −5.62993930176573612868681603999, −5.40073527479789986487080806539, −5.35145631153365257257288427174, −5.18326799711102433496251762636, −4.50899027421435645999146800481, −4.47484772380230149505242221305, −3.89559839357338646144793355676, −3.74026968444689358944525993793, −3.64930127054741408285637396511, −3.47503266096435616241455344497, −2.96955458023403756790430587966, −2.65901321477146652979215573245, −2.57176184614156839015662782998, −2.51326560655555123499681572560, −1.85544647224117730049348437037, −1.39632213943218962268505421934, −1.11057177852346342723139162784, −0.967066258156891092513830663489, −0.35891948470655134718687340536,
0.35891948470655134718687340536, 0.967066258156891092513830663489, 1.11057177852346342723139162784, 1.39632213943218962268505421934, 1.85544647224117730049348437037, 2.51326560655555123499681572560, 2.57176184614156839015662782998, 2.65901321477146652979215573245, 2.96955458023403756790430587966, 3.47503266096435616241455344497, 3.64930127054741408285637396511, 3.74026968444689358944525993793, 3.89559839357338646144793355676, 4.47484772380230149505242221305, 4.50899027421435645999146800481, 5.18326799711102433496251762636, 5.35145631153365257257288427174, 5.40073527479789986487080806539, 5.62993930176573612868681603999, 6.14218864974407392950114274744, 6.33146585074858009467525168055, 6.38360144564905115530464059347, 6.44812847748733345634131191308, 7.35065036165764919889751796892, 7.41658315979107512340847711271
