| L(s) = 1 | − 2·9-s + 8·13-s − 4·25-s − 8·37-s + 36·49-s + 8·61-s + 2·81-s − 48·97-s + 40·109-s − 16·117-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.21·13-s − 4/5·25-s − 1.31·37-s + 36/7·49-s + 1.02·61-s + 2/9·81-s − 4.87·97-s + 3.83·109-s − 1.47·117-s − 4.36·121-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 + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.996229699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996229699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| good | 7 | \( ( 1 - 18 T^{2} + 162 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 - 56 T^{2} + 1438 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 34 T^{2} + 514 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 80 T^{2} + 3214 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 96 T^{2} + 4158 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 110 T^{2} + 5890 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 178 T^{2} + 12322 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 44 T^{2} + 1750 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 16 T^{2} + 1518 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 222 T^{2} + 21282 T^{4} - 222 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 228 T^{2} + 22806 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 168 T^{2} + 19470 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 82 T^{2} + 4834 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 212 T^{2} + 25990 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
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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
−4.18890289571982489832558064048, −4.12326062997423796001799214377, −4.07897659069863168554349251843, −3.90350038298692946043145910251, −3.75024211311188530138845145294, −3.60362251330781105921286427909, −3.59895468519661589364563265603, −3.57754742081845160691364355775, −3.17699623539425017319657930453, −3.13867065856661612159732177044, −2.85038785527138762740415210921, −2.75319910985165030082078448235, −2.69135329919488648704785523185, −2.44634767420328228480978451930, −2.36431209826502088299901049207, −2.33277974300831432861263484361, −1.97298467338600515819011028300, −1.67414901358334986745375728106, −1.65522069524231165117366213529, −1.52509345949167648296592625454, −1.26488667624868477988341166564, −1.01909066393524675857853061653, −0.77590398662705480563984510010, −0.63019049406000942639412338643, −0.21040371494137821467693618528,
0.21040371494137821467693618528, 0.63019049406000942639412338643, 0.77590398662705480563984510010, 1.01909066393524675857853061653, 1.26488667624868477988341166564, 1.52509345949167648296592625454, 1.65522069524231165117366213529, 1.67414901358334986745375728106, 1.97298467338600515819011028300, 2.33277974300831432861263484361, 2.36431209826502088299901049207, 2.44634767420328228480978451930, 2.69135329919488648704785523185, 2.75319910985165030082078448235, 2.85038785527138762740415210921, 3.13867065856661612159732177044, 3.17699623539425017319657930453, 3.57754742081845160691364355775, 3.59895468519661589364563265603, 3.60362251330781105921286427909, 3.75024211311188530138845145294, 3.90350038298692946043145910251, 4.07897659069863168554349251843, 4.12326062997423796001799214377, 4.18890289571982489832558064048
Plot not available for L-functions of degree greater than 10.
