| L(s) = 1 | − 2·4-s + 4·5-s + 3·16-s − 16·19-s − 8·20-s + 8·25-s − 8·29-s + 16·31-s − 24·41-s − 2·49-s + 16·59-s + 24·61-s − 4·64-s − 24·71-s + 32·76-s − 8·79-s + 12·80-s − 18·81-s + 40·89-s − 64·95-s − 16·100-s + 24·101-s − 8·109-s + 16·116-s + 4·121-s − 32·124-s + 20·125-s + ⋯ |
| L(s) = 1 | − 4-s + 1.78·5-s + 3/4·16-s − 3.67·19-s − 1.78·20-s + 8/5·25-s − 1.48·29-s + 2.87·31-s − 3.74·41-s − 2/7·49-s + 2.08·59-s + 3.07·61-s − 1/2·64-s − 2.84·71-s + 3.67·76-s − 0.900·79-s + 1.34·80-s − 2·81-s + 4.23·89-s − 6.56·95-s − 8/5·100-s + 2.38·101-s − 0.766·109-s + 1.48·116-s + 4/11·121-s − 2.87·124-s + 1.78·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6998201517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6998201517\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−10.73426123510157600035228413424, −10.43134582684158402341125059784, −10.20015222730524925012267609842, −10.14426114234254611513281680593, −9.972584520176235140764895497419, −9.522611812166983285514215208849, −9.099332830909959217552901262909, −8.648810880927592417393614859814, −8.555439577994667383484011730629, −8.513163680692904578369236277240, −8.163324242375110243183970953295, −7.41215864755701840767877718374, −7.16680802511913941244908265704, −6.44253967075169279419816373286, −6.41637525834994849944410655192, −6.31546491003053097024332362523, −5.69448156402581075408846634244, −5.29435469974098243385809056628, −4.92316433075312663206256433053, −4.61801799198574011265464084909, −4.04853735820217459631622764883, −3.74623558901870409569966711662, −2.90486681587310966934221475522, −2.20761612620777380177549525069, −1.82896760108220512789219906716,
1.82896760108220512789219906716, 2.20761612620777380177549525069, 2.90486681587310966934221475522, 3.74623558901870409569966711662, 4.04853735820217459631622764883, 4.61801799198574011265464084909, 4.92316433075312663206256433053, 5.29435469974098243385809056628, 5.69448156402581075408846634244, 6.31546491003053097024332362523, 6.41637525834994849944410655192, 6.44253967075169279419816373286, 7.16680802511913941244908265704, 7.41215864755701840767877718374, 8.163324242375110243183970953295, 8.513163680692904578369236277240, 8.555439577994667383484011730629, 8.648810880927592417393614859814, 9.099332830909959217552901262909, 9.522611812166983285514215208849, 9.972584520176235140764895497419, 10.14426114234254611513281680593, 10.20015222730524925012267609842, 10.43134582684158402341125059784, 10.73426123510157600035228413424
