L(s) = 1 | + 48·13-s − 8·25-s − 4·49-s + 48·61-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.19e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 13.3·13-s − 8/5·25-s − 4/7·49-s + 6.14·61-s + 8/11·121-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 + 91.6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(119.6902538\) |
\(L(\frac12)\) |
\(\approx\) |
\(119.6902538\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
good | 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
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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
−3.65705471080018795163061822498, −3.35558026292893268864619091948, −3.30922860052487716512540987952, −3.22852104590646133007663059317, −3.15984377718022982854165654482, −3.08749187990076293728478503331, −2.78852668431836488478589841754, −2.73719554675357095856691357436, −2.71778572846760938308815414933, −2.44848134348943119606803756226, −2.14639057490309589276468662972, −1.91571470477267452135372930172, −1.87078273906122144932137083950, −1.79459902228397515312269924589, −1.65475783350659674332469930151, −1.64509798484352960354076600543, −1.55713160420239701321147716183, −1.47742321527226002282585392981, −1.09758447885946931964007692419, −0.921576956845613378767616426951, −0.914005455695813298752432807034, −0.827658362095883968433396681585, −0.78332386986957885637710163276, −0.70724620021290127199506607613, −0.34812622833819661908514973242,
0.34812622833819661908514973242, 0.70724620021290127199506607613, 0.78332386986957885637710163276, 0.827658362095883968433396681585, 0.914005455695813298752432807034, 0.921576956845613378767616426951, 1.09758447885946931964007692419, 1.47742321527226002282585392981, 1.55713160420239701321147716183, 1.64509798484352960354076600543, 1.65475783350659674332469930151, 1.79459902228397515312269924589, 1.87078273906122144932137083950, 1.91571470477267452135372930172, 2.14639057490309589276468662972, 2.44848134348943119606803756226, 2.71778572846760938308815414933, 2.73719554675357095856691357436, 2.78852668431836488478589841754, 3.08749187990076293728478503331, 3.15984377718022982854165654482, 3.22852104590646133007663059317, 3.30922860052487716512540987952, 3.35558026292893268864619091948, 3.65705471080018795163061822498
Plot not available for L-functions of degree greater than 10.