| L(s) = 1 | − 4·11-s + 16-s + 4·19-s + 48·31-s − 16·59-s + 36·61-s + 8·71-s + 84·79-s + 81-s − 12·89-s − 36·101-s + 60·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 1/4·16-s + 0.917·19-s + 8.62·31-s − 2.08·59-s + 4.60·61-s + 0.949·71-s + 9.45·79-s + 1/9·81-s − 1.27·89-s − 3.58·101-s + 5.74·109-s + 4·121-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 + 0.0773·167-s + 0.0760·173-s − 0.301·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.58741250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.58741250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 71 T^{4} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 92 T^{4} - 17562 T^{8} - 92 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2}( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 19 | \( ( 1 - 2 T - 8 T^{2} + 52 T^{3} - 293 T^{4} + 52 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 24 T^{2} - 223 T^{4} + 9960 T^{6} - 6048 T^{8} + 9960 p^{2} T^{10} - 223 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 44 T^{2} + 1194 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 24 T + 301 T^{2} - 2616 T^{3} + 16872 T^{4} - 2616 p T^{5} + 301 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 36 T^{2} - 566 T^{4} - 35928 T^{6} - 286749 T^{8} - 35928 p^{2} T^{10} - 566 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 138 T^{2} + 8075 T^{4} - 138 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 316 T^{4} - 5622234 T^{8} + 316 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 3689 T^{4} + 8729040 T^{8} + 3689 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 - 84 T^{2} + 5642 T^{4} - 276360 T^{6} + 9540387 T^{8} - 276360 p^{2} T^{10} + 5642 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 8 T - 58 T^{2} + 32 T^{3} + 7627 T^{4} + 32 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 18 T + 208 T^{2} - 1800 T^{3} + 12867 T^{4} - 1800 p T^{5} + 208 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 4946 T^{4} + 4311795 T^{8} - 4946 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 3934 T^{4} - 12921885 T^{8} + 3934 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 42 T + 877 T^{2} - 12138 T^{3} + 123732 T^{4} - 12138 p T^{5} + 877 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 8708 T^{4} + 41254758 T^{8} - 8708 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 6 T - 139 T^{2} - 18 T^{3} + 19500 T^{4} - 18 p T^{5} - 139 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 2207 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.34889285311561294158859094696, −4.11782526345850370233484842059, −4.02013013325127401869894072092, −3.81018234974733768556240831367, −3.71835566254617654308353305387, −3.49236267259999509866646754814, −3.40937965742944531405486553512, −3.25919435667247605892374069994, −3.25183386745905827677024407660, −3.00695804995083746580738795388, −2.98013405053856835427705152076, −2.64423503146944905601645445522, −2.51627578151184366924147848763, −2.42442961499939216834761930595, −2.31772546583076593157256743508, −2.23668431639337899672787761052, −2.19688598508023979790749865317, −1.87333421162344474701400810406, −1.63576482992958249204711915600, −1.09198388319321500295533909063, −1.05981417904180312485656760289, −0.979197188777169962275435093680, −0.975349248856731245457892670401, −0.57928237720953143500420945154, −0.43166284668963841133879670993,
0.43166284668963841133879670993, 0.57928237720953143500420945154, 0.975349248856731245457892670401, 0.979197188777169962275435093680, 1.05981417904180312485656760289, 1.09198388319321500295533909063, 1.63576482992958249204711915600, 1.87333421162344474701400810406, 2.19688598508023979790749865317, 2.23668431639337899672787761052, 2.31772546583076593157256743508, 2.42442961499939216834761930595, 2.51627578151184366924147848763, 2.64423503146944905601645445522, 2.98013405053856835427705152076, 3.00695804995083746580738795388, 3.25183386745905827677024407660, 3.25919435667247605892374069994, 3.40937965742944531405486553512, 3.49236267259999509866646754814, 3.71835566254617654308353305387, 3.81018234974733768556240831367, 4.02013013325127401869894072092, 4.11782526345850370233484842059, 4.34889285311561294158859094696
Plot not available for L-functions of degree greater than 10.
