| L(s) = 1 | − 8·3-s + 32·9-s + 8·13-s − 2·16-s − 20·25-s − 88·27-s + 32·31-s + 16·37-s − 64·39-s − 48·43-s + 16·48-s + 64·61-s + 16·67-s − 8·73-s + 160·75-s + 206·81-s − 256·93-s − 16·97-s − 64·103-s − 128·111-s + 256·117-s + 56·121-s + 127-s + 384·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 32/3·9-s + 2.21·13-s − 1/2·16-s − 4·25-s − 16.9·27-s + 5.74·31-s + 2.63·37-s − 10.2·39-s − 7.31·43-s + 2.30·48-s + 8.19·61-s + 1.95·67-s − 0.936·73-s + 18.4·75-s + 22.8·81-s − 26.5·93-s − 1.62·97-s − 6.30·103-s − 12.1·111-s + 23.6·117-s + 5.09·121-s + 0.0887·127-s + 33.8·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3073858926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3073858926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + p T^{2} )^{4} \) |
| 23 | \( ( 1 + T^{4} )^{2} \) |
| good | 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 13 | \( ( 1 - 4 T + 8 T^{2} + 20 T^{3} - 274 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 36 T^{4} - 88634 T^{8} - 36 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 48 T^{2} + 1138 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 32 T^{2} - 200 T^{3} + 1106 T^{4} - 200 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 12 T^{2} + 838 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 24 T + 288 T^{2} + 2280 T^{3} + 15346 T^{4} + 2280 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 3332 T^{4} + 5069958 T^{8} + 3332 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 5362 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 - 8 T + 32 T^{2} - 440 T^{3} + 5906 T^{4} - 440 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 240 T^{2} + 24322 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 8 T^{2} - 20 T^{3} - 6034 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 148 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 3492 T^{4} + 59862118 T^{8} + 3492 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 52 T^{2} - 6522 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 8 T + 32 T^{2} + 680 T^{3} + 14306 T^{4} + 680 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.65728336252795099705528580465, −4.33099613638906673084004542123, −4.24095318563335927402088242370, −4.21148002782277551353241201141, −4.13733923143049356280263859201, −3.98192286443537433050914471755, −3.85035966592552600181741963429, −3.83423178494857693586391526688, −3.65766338503335376828317630412, −3.23993514281653328672569629859, −3.10980569070032016261842038759, −3.00489455149613466222323655977, −2.88662618544675750812313032048, −2.77402710710490542156853599727, −2.28486817663019095808799361808, −2.18574697193643455854648703239, −2.07113355285060370684453634066, −1.73710811464688145499964739068, −1.65352922810162116145508389694, −1.62283979579580855374254019429, −0.958778701376155870771859703289, −0.884448797860282078120196194643, −0.872111323977647044875938444791, −0.59463492846513305198195169951, −0.19384124299261953454612110680,
0.19384124299261953454612110680, 0.59463492846513305198195169951, 0.872111323977647044875938444791, 0.884448797860282078120196194643, 0.958778701376155870771859703289, 1.62283979579580855374254019429, 1.65352922810162116145508389694, 1.73710811464688145499964739068, 2.07113355285060370684453634066, 2.18574697193643455854648703239, 2.28486817663019095808799361808, 2.77402710710490542156853599727, 2.88662618544675750812313032048, 3.00489455149613466222323655977, 3.10980569070032016261842038759, 3.23993514281653328672569629859, 3.65766338503335376828317630412, 3.83423178494857693586391526688, 3.85035966592552600181741963429, 3.98192286443537433050914471755, 4.13733923143049356280263859201, 4.21148002782277551353241201141, 4.24095318563335927402088242370, 4.33099613638906673084004542123, 4.65728336252795099705528580465
Plot not available for L-functions of degree greater than 10.
