| L(s) = 1 | + 8·2-s + 32·4-s + 80·8-s + 120·16-s − 24·23-s + 12·25-s + 32·32-s − 192·46-s + 96·50-s − 384·64-s + 32·71-s − 768·92-s + 384·100-s + 88·121-s + 127-s − 1.21e3·128-s + 131-s + 137-s + 139-s + 256·142-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 16·4-s + 28.2·8-s + 30·16-s − 5.00·23-s + 12/5·25-s + 5.65·32-s − 28.3·46-s + 13.5·50-s − 48·64-s + 3.79·71-s − 80.0·92-s + 38.3·100-s + 8·121-s + 0.0887·127-s − 107.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 21.4·142-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.78908088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.78908088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T + p T^{2} )^{4} \) |
| 5 | \( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 8 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )( 1 + 4 T^{2} + 8 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 17 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 12 T^{2} + 72 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 6 T + p T^{2} )^{4}( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 108 T^{2} + 5832 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 12 T^{2} + 72 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 + p^{2} T^{4} )^{4} \) |
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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
−5.37307288625315374445527250212, −5.15208992771478852450653058733, −5.08513957121305046705508251564, −4.85483556443704950444838208912, −4.54041860947147154718760251169, −4.47068490601417433318671083236, −4.41053986502618376954193049358, −4.34093375767852753195323479610, −4.30285224622664934297551790744, −4.04075786325345303723552738706, −3.88464178802864069036729086043, −3.64150647869528983291310787503, −3.43386252445853009353079482271, −3.32265975946474870694694836611, −3.29633260994323964361219565220, −3.16868476409146693255387797604, −3.10049946236278568573381957593, −2.50078484901365260922877867185, −2.48052472986942255065423867360, −2.19328723600504689848836826335, −2.16704041827730381021881526940, −1.95294742844658589481979869451, −1.92617883642281988941808591821, −1.12240344117003200361874487833, −0.43010155017251591624800212367,
0.43010155017251591624800212367, 1.12240344117003200361874487833, 1.92617883642281988941808591821, 1.95294742844658589481979869451, 2.16704041827730381021881526940, 2.19328723600504689848836826335, 2.48052472986942255065423867360, 2.50078484901365260922877867185, 3.10049946236278568573381957593, 3.16868476409146693255387797604, 3.29633260994323964361219565220, 3.32265975946474870694694836611, 3.43386252445853009353079482271, 3.64150647869528983291310787503, 3.88464178802864069036729086043, 4.04075786325345303723552738706, 4.30285224622664934297551790744, 4.34093375767852753195323479610, 4.41053986502618376954193049358, 4.47068490601417433318671083236, 4.54041860947147154718760251169, 4.85483556443704950444838208912, 5.08513957121305046705508251564, 5.15208992771478852450653058733, 5.37307288625315374445527250212
Plot not available for L-functions of degree greater than 10.
