| L(s) = 1 | + 4·7-s + 6·13-s + 12·17-s − 16·19-s − 8·25-s − 2·43-s + 12·47-s − 6·49-s − 24·59-s + 2·61-s + 6·67-s − 12·71-s + 14·73-s − 18·79-s + 24·91-s − 36·97-s − 36·101-s − 24·107-s + 48·119-s + 4·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.66·13-s + 2.91·17-s − 3.67·19-s − 8/5·25-s − 0.304·43-s + 1.75·47-s − 6/7·49-s − 3.12·59-s + 0.256·61-s + 0.733·67-s − 1.42·71-s + 1.63·73-s − 2.02·79-s + 2.51·91-s − 3.65·97-s − 3.58·101-s − 2.32·107-s + 4.40·119-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.221399333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.221399333\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2463 T^{4} - 528 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 59 T^{2} - 46 T^{3} + 1948 T^{4} - 46 p T^{5} - 59 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 152 T^{2} - 1248 T^{3} + 10863 T^{4} - 1248 p T^{5} + 152 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 320 T^{2} + 3312 T^{3} + 28071 T^{4} + 3312 p T^{5} + 320 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 390 T^{3} + 540 T^{4} - 390 p T^{5} + 77 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 275 T^{2} + 3006 T^{3} + 30180 T^{4} + 3006 p T^{5} + 275 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.10763195428405817302696192878, −6.10155366383122081620147513479, −5.85091007390287389818891282118, −5.56030380873662557644912392553, −5.55030930933979473501242946355, −5.36880355583514362488468680838, −5.00745894211047923975235024604, −4.81226828671111579549847094280, −4.53292551631563354012396507438, −4.32512054519750709689735791869, −4.08562092367095796951506087054, −3.98931447689414135389275425362, −3.80260676877354525008802544650, −3.71469928791547836973468570659, −3.20165926242682729976985538956, −2.85211178104411688952897683970, −2.84718886001384022433994239774, −2.64571557677682323745443393841, −1.99781584331197555497295916155, −1.96230387895430539692664771612, −1.46362198568462119419152114756, −1.44449615105317904145381037266, −1.42424820199624414917069757087, −0.74620476490508314624978228702, −0.16438642888878608902395100059,
0.16438642888878608902395100059, 0.74620476490508314624978228702, 1.42424820199624414917069757087, 1.44449615105317904145381037266, 1.46362198568462119419152114756, 1.96230387895430539692664771612, 1.99781584331197555497295916155, 2.64571557677682323745443393841, 2.84718886001384022433994239774, 2.85211178104411688952897683970, 3.20165926242682729976985538956, 3.71469928791547836973468570659, 3.80260676877354525008802544650, 3.98931447689414135389275425362, 4.08562092367095796951506087054, 4.32512054519750709689735791869, 4.53292551631563354012396507438, 4.81226828671111579549847094280, 5.00745894211047923975235024604, 5.36880355583514362488468680838, 5.55030930933979473501242946355, 5.56030380873662557644912392553, 5.85091007390287389818891282118, 6.10155366383122081620147513479, 6.10763195428405817302696192878
