| L(s) = 1 | + 2·2-s + 5·4-s − 8·7-s + 4·8-s − 2·9-s − 16·14-s + 4·16-s − 4·18-s + 8·19-s − 40·25-s − 40·28-s − 22·32-s − 10·36-s + 16·38-s + 4·41-s − 24·47-s + 30·49-s − 80·50-s − 32·56-s − 16·59-s + 16·63-s − 44·64-s − 32·67-s − 16·71-s − 8·72-s + 40·76-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 5/2·4-s − 3.02·7-s + 1.41·8-s − 2/3·9-s − 4.27·14-s + 16-s − 0.942·18-s + 1.83·19-s − 8·25-s − 7.55·28-s − 3.88·32-s − 5/3·36-s + 2.59·38-s + 0.624·41-s − 3.50·47-s + 30/7·49-s − 11.3·50-s − 4.27·56-s − 2.08·59-s + 2.01·63-s − 5.5·64-s − 3.90·67-s − 1.89·71-s − 0.942·72-s + 4.58·76-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5435851090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5435851090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + 2 T^{2} - 5 T^{4} - 28 T^{6} - 11 T^{8} - 28 p^{2} T^{10} - 5 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 + p T^{2} )^{8} \) |
| 7 | \( ( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} + 8 p T^{5} + 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 14 T^{2} + 75 T^{4} + 644 T^{6} - 18091 T^{8} + 644 p^{2} T^{10} + 75 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 24 T^{2} + 407 T^{4} - 5712 T^{6} + 68305 T^{8} - 5712 p^{2} T^{10} + 407 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 32 T^{2} + 735 T^{4} - 14272 T^{6} + 244289 T^{8} - 14272 p^{2} T^{10} + 735 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 4 T - 3 T^{2} + 88 T^{3} - 295 T^{4} + 88 p T^{5} - 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 14 T^{2} - 333 T^{4} + 12068 T^{6} + 7205 T^{8} + 12068 p^{2} T^{10} - 333 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 - 56 T^{2} + 2295 T^{4} - 81424 T^{6} + 2629649 T^{8} - 81424 p^{2} T^{10} + 2295 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 156 p T^{5} - 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 14 T^{2} - 1653 T^{4} + 49028 T^{6} + 2370005 T^{8} + 49028 p^{2} T^{10} - 1653 p^{4} T^{12} - 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 600 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 88 T^{2} + 4935 T^{4} - 187088 T^{6} + 2601329 T^{8} - 187088 p^{2} T^{10} + 4935 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 8 T + 5 T^{2} - 432 T^{3} - 3751 T^{4} - 432 p T^{5} + 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 8 T - 7 T^{2} - 624 T^{3} - 4495 T^{4} - 624 p T^{5} - 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 128 T^{2} + 11055 T^{4} - 732928 T^{6} + 34902689 T^{8} - 732928 p^{2} T^{10} + 11055 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 30 T^{2} - 5341 T^{4} + 347460 T^{6} + 22909381 T^{8} + 347460 p^{2} T^{10} - 5341 p^{4} T^{12} - 30 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 + 34 T^{2} - 5733 T^{4} - 429148 T^{6} + 24903605 T^{8} - 429148 p^{2} T^{10} - 5733 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 128 T^{2} + 8463 T^{4} - 69376 T^{6} - 58155295 T^{8} - 69376 p^{2} T^{10} + 8463 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 8 T - 33 T^{2} - 1040 T^{3} - 5119 T^{4} - 1040 p T^{5} - 33 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.20920402739178094077432048899, −4.10264067400641663459788834061, −4.00613070358330865625264534731, −3.85189044362347456404444368818, −3.64339140380163334650393725983, −3.49372927019269652497706819915, −3.38895091248599708654057576514, −3.38012084922844866881492975228, −3.30749823697700474063424828720, −3.30541539917666214436583216260, −3.08903106456124521169085803083, −2.80101347493079498441657035596, −2.67538618661698730870904822945, −2.61233695252139389758908225888, −2.45206236445472566432441286515, −2.31867655238823765203183921213, −1.89654529688565128923031207327, −1.87932386944512873731836542569, −1.71691447868741558980655395382, −1.71376050206020056652030416325, −1.46741319083444709556948620869, −1.31113221548804069886282760625, −0.42069047049783314087943704927, −0.34733634611187430549729242984, −0.15113960954661166720935292978,
0.15113960954661166720935292978, 0.34733634611187430549729242984, 0.42069047049783314087943704927, 1.31113221548804069886282760625, 1.46741319083444709556948620869, 1.71376050206020056652030416325, 1.71691447868741558980655395382, 1.87932386944512873731836542569, 1.89654529688565128923031207327, 2.31867655238823765203183921213, 2.45206236445472566432441286515, 2.61233695252139389758908225888, 2.67538618661698730870904822945, 2.80101347493079498441657035596, 3.08903106456124521169085803083, 3.30541539917666214436583216260, 3.30749823697700474063424828720, 3.38012084922844866881492975228, 3.38895091248599708654057576514, 3.49372927019269652497706819915, 3.64339140380163334650393725983, 3.85189044362347456404444368818, 4.00613070358330865625264534731, 4.10264067400641663459788834061, 4.20920402739178094077432048899
Plot not available for L-functions of degree greater than 10.
