L(s) = 1 | − 4·4-s + 4·16-s − 16·19-s − 16·41-s + 64·59-s + 24·61-s + 16·64-s + 64·76-s − 32·79-s − 2·81-s + 16·101-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 64·164-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 2·4-s + 16-s − 3.67·19-s − 2.49·41-s + 8.33·59-s + 3.07·61-s + 2·64-s + 7.34·76-s − 3.60·79-s − 2/9·81-s + 1.59·101-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 4.99·164-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05940581789\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05940581789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2 p^{2} T^{4} + 7011 T^{8} - 2 p^{6} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 194 T^{4} + 171 p^{2} T^{8} - 194 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 68 T^{4} - 434490 T^{8} + 68 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 60 T^{2} + 1814 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 110 T^{2} + 4899 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 868 T^{4} + 2707878 T^{8} + 868 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 3358 T^{4} + 8633475 T^{8} + 3358 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 5188 T^{4} + 13723398 T^{8} + 5188 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 + 5348 T^{4} + 16710438 T^{8} + 5348 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 67 | \( 1 - 6818 T^{4} + 25780611 T^{8} - 6818 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 188 T^{2} + 17190 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 6242 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 - 284 T^{4} - 90968346 T^{8} - 284 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 3842 T^{4} + 131200515 T^{8} - 3842 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−5.23688680675532335963240735048, −5.03934894693973852684617323264, −4.96095121256671559131796653808, −4.93870917422369993105520452417, −4.70488089812656058169457790238, −4.47427741858996139811691191966, −4.34424633014078671231037675409, −4.10618377130692372675998514176, −4.01907419273732841686654262399, −3.84293031434998627054576700185, −3.82709420516811561868415359340, −3.75480238412235295769257228767, −3.67359156283104065706766136618, −3.20509406619514771883719907166, −3.00907501542610901833139583488, −2.88190481429644464415862948045, −2.33696543314545808332484879466, −2.31723318336624174704414250190, −2.25556464027707890346442398374, −2.19322927879178443119481888956, −1.92753299745180275473370532683, −1.40934423812448630895860693736, −1.01698656125649746813744628889, −0.803625331949987604251320135723, −0.085887207855476897652543601301,
0.085887207855476897652543601301, 0.803625331949987604251320135723, 1.01698656125649746813744628889, 1.40934423812448630895860693736, 1.92753299745180275473370532683, 2.19322927879178443119481888956, 2.25556464027707890346442398374, 2.31723318336624174704414250190, 2.33696543314545808332484879466, 2.88190481429644464415862948045, 3.00907501542610901833139583488, 3.20509406619514771883719907166, 3.67359156283104065706766136618, 3.75480238412235295769257228767, 3.82709420516811561868415359340, 3.84293031434998627054576700185, 4.01907419273732841686654262399, 4.10618377130692372675998514176, 4.34424633014078671231037675409, 4.47427741858996139811691191966, 4.70488089812656058169457790238, 4.93870917422369993105520452417, 4.96095121256671559131796653808, 5.03934894693973852684617323264, 5.23688680675532335963240735048
Plot not available for L-functions of degree greater than 10.