| L(s) = 1 | + 4·3-s − 4·4-s − 2·9-s − 16·12-s + 10·16-s − 8·19-s − 4·25-s − 40·27-s + 8·36-s + 16·37-s + 40·48-s − 12·49-s − 32·57-s + 8·61-s − 20·64-s + 56·67-s − 16·75-s + 32·76-s − 55·81-s + 80·97-s + 16·100-s + 8·103-s + 160·108-s + 64·111-s + 40·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 2·4-s − 2/3·9-s − 4.61·12-s + 5/2·16-s − 1.83·19-s − 4/5·25-s − 7.69·27-s + 4/3·36-s + 2.63·37-s + 5.77·48-s − 1.71·49-s − 4.23·57-s + 1.02·61-s − 5/2·64-s + 6.84·67-s − 1.84·75-s + 3.67·76-s − 6.11·81-s + 8.12·97-s + 8/5·100-s + 0.788·103-s + 15.3·108-s + 6.07·111-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.001465030511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001465030511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( ( 1 - T + p T^{2} )^{4} \) |
| 5 | \( 1 + 4 T^{2} + 2 p T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 6 T^{2} + 9 p T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 298 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 2 p T^{2} + 823 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 6 T^{2} + 1023 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 46 T^{2} + 1111 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 44 T^{2} + 1306 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 20 T^{2} - 102 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 156 T^{2} + 9738 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 84 T^{2} + 5478 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 105 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 106 T^{2} + 8671 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2 T + 112 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 7 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 60 T^{2} + 10938 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 166 T^{2} + 13983 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 60 T^{2} + 2118 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 196 T^{2} + 22678 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 276 T^{2} + 33786 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 20 T + 250 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.84125416569397465383804606521, −4.46052939774307538651339336658, −4.34830857532347625125336907580, −4.28048144311550504357408297732, −3.97487436866987040732144443145, −3.80826456301963973685822028475, −3.79552961678955395185894640897, −3.68641635899840647004933818209, −3.54757198265608738185286201931, −3.54241001469395644329985393235, −3.40399614788660971277999040548, −3.05574370322114375271676451343, −2.95117910463557214406218736876, −2.78343948186860015235716336647, −2.59831180681730077909049536057, −2.34738329700134853225669742141, −2.26510669008699351685206584032, −2.18169764214780526648115895234, −2.08538339341890523988858211320, −2.02166679814079099484046829008, −1.47048845668974070536271096943, −1.10166048778208618671387607696, −0.874620487700127789595186109791, −0.60924554543560843036988153323, −0.00694900430621809062069547377,
0.00694900430621809062069547377, 0.60924554543560843036988153323, 0.874620487700127789595186109791, 1.10166048778208618671387607696, 1.47048845668974070536271096943, 2.02166679814079099484046829008, 2.08538339341890523988858211320, 2.18169764214780526648115895234, 2.26510669008699351685206584032, 2.34738329700134853225669742141, 2.59831180681730077909049536057, 2.78343948186860015235716336647, 2.95117910463557214406218736876, 3.05574370322114375271676451343, 3.40399614788660971277999040548, 3.54241001469395644329985393235, 3.54757198265608738185286201931, 3.68641635899840647004933818209, 3.79552961678955395185894640897, 3.80826456301963973685822028475, 3.97487436866987040732144443145, 4.28048144311550504357408297732, 4.34830857532347625125336907580, 4.46052939774307538651339336658, 4.84125416569397465383804606521
Plot not available for L-functions of degree greater than 10.
