| L(s) = 1 | − 4·2-s + 6·4-s + 8·11-s − 15·16-s − 32·22-s − 8·23-s + 4·25-s + 16·29-s + 24·32-s + 32·37-s − 8·43-s + 48·44-s + 32·46-s − 16·50-s + 32·53-s − 64·58-s − 6·64-s − 36·67-s + 48·71-s − 128·74-s − 8·79-s + 9·81-s + 32·86-s − 48·92-s + 24·100-s − 128·106-s − 56·107-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s + 2.41·11-s − 3.75·16-s − 6.82·22-s − 1.66·23-s + 4/5·25-s + 2.97·29-s + 4.24·32-s + 5.26·37-s − 1.21·43-s + 7.23·44-s + 4.71·46-s − 2.26·50-s + 4.39·53-s − 8.40·58-s − 3/4·64-s − 4.39·67-s + 5.69·71-s − 14.8·74-s − 0.900·79-s + 81-s + 3.45·86-s − 5.00·92-s + 12/5·100-s − 12.4·106-s − 5.41·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.435824914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435824914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T^{2} + 2 p T^{4} + 176 T^{6} - 989 T^{8} + 176 p^{2} T^{10} + 2 p^{5} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 4 T - 7 T^{2} - 4 T^{3} + 232 T^{4} - 4 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 4 T^{2} + 106 T^{4} + 1712 T^{6} - 23165 T^{8} + 1712 p^{2} T^{10} + 106 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T^{2} + 519 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1271 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 4 T - 22 T^{2} - 32 T^{3} + 547 T^{4} - 32 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 76 T^{2} + 2842 T^{4} - 76912 T^{6} + 2134099 T^{8} - 76912 p^{2} T^{10} + 2842 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 88 T^{2} + 3769 T^{4} - 53944 T^{6} - 42800 T^{8} - 53944 p^{2} T^{10} + 3769 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 4 T - 71 T^{2} + 4 T^{3} + 5032 T^{4} + 4 p T^{5} - 71 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 76 T^{2} + 2986 T^{4} + 123728 T^{6} - 9854765 T^{8} + 123728 p^{2} T^{10} + 2986 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 160 T^{2} + 13561 T^{4} - 812320 T^{6} + 43706560 T^{8} - 812320 p^{2} T^{10} + 13561 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 36 T^{2} - 3398 T^{4} + 98928 T^{6} + 3365379 T^{8} + 98928 p^{2} T^{10} - 3398 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 18 T + 121 T^{2} + 1242 T^{3} + 15012 T^{4} + 1242 p T^{5} + 121 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 216 T^{2} + 22247 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 4 T - 98 T^{2} - 176 T^{3} + 5491 T^{4} - 176 p T^{5} - 98 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 32 T^{2} + 6190 T^{4} - 606208 T^{6} - 27825101 T^{8} - 606208 p^{2} T^{10} + 6190 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 24 T^{2} + 3289 T^{4} + 516744 T^{6} - 86588496 T^{8} + 516744 p^{2} T^{10} + 3289 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.30036268262124481246441346057, −4.20193042108475748739791064214, −4.12950857176635355555978626560, −4.05931141604653588582484377017, −3.87900864798350283206614407965, −3.83187980918785329076372693431, −3.49157253478513465921269975493, −3.36411679711814284731639448354, −3.27269775823757453843459438636, −3.16421136699353044390520487107, −2.83137244861070988870887646962, −2.77929783830208376671381606112, −2.46685769890705190774969454148, −2.41172285243084854084617369788, −2.26844306103258841254698583377, −2.20118106182303858975261166433, −2.02604585211529607581634180174, −1.67758511363115581594255527495, −1.42532455514071576235778048113, −1.20894586477907129115251380887, −1.13459631287917257943970066423, −1.02627070929023879854352178936, −0.870878447776172000689422353075, −0.54861880055944635791423839209, −0.35882942792989558673069699371,
0.35882942792989558673069699371, 0.54861880055944635791423839209, 0.870878447776172000689422353075, 1.02627070929023879854352178936, 1.13459631287917257943970066423, 1.20894586477907129115251380887, 1.42532455514071576235778048113, 1.67758511363115581594255527495, 2.02604585211529607581634180174, 2.20118106182303858975261166433, 2.26844306103258841254698583377, 2.41172285243084854084617369788, 2.46685769890705190774969454148, 2.77929783830208376671381606112, 2.83137244861070988870887646962, 3.16421136699353044390520487107, 3.27269775823757453843459438636, 3.36411679711814284731639448354, 3.49157253478513465921269975493, 3.83187980918785329076372693431, 3.87900864798350283206614407965, 4.05931141604653588582484377017, 4.12950857176635355555978626560, 4.20193042108475748739791064214, 4.30036268262124481246441346057
Plot not available for L-functions of degree greater than 10.
