| L(s) = 1 | + 6·4-s − 8·5-s + 2·9-s + 19·16-s − 48·20-s + 36·25-s + 12·36-s + 4·37-s − 16·45-s − 34·49-s − 36·53-s + 36·64-s − 152·80-s + 19·81-s − 28·89-s + 16·97-s + 216·100-s + 80·113-s − 120·125-s + 127-s + 131-s + 137-s + 139-s + 38·144-s + 24·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3·4-s − 3.57·5-s + 2/3·9-s + 19/4·16-s − 10.7·20-s + 36/5·25-s + 2·36-s + 0.657·37-s − 2.38·45-s − 4.85·49-s − 4.94·53-s + 9/2·64-s − 16.9·80-s + 19/9·81-s − 2.96·89-s + 1.62·97-s + 21.5·100-s + 7.52·113-s − 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 19/6·144-s + 1.97·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.573748054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573748054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + T )^{8} \) |
| 11 | \( 1 - 178 T^{4} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + 17 T^{2} + 144 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 15 T^{2} + 608 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 5 T^{2} + 492 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 48 T^{2} + 1214 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 51 T^{2} + 1676 T^{4} - 51 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 113 T^{2} + 5088 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | \( ( 1 + 128 T^{2} + 7374 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 144 T^{2} + 9182 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 80 T^{2} + 4782 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 107 T^{2} + 5868 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 185 T^{2} + 16512 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 140 T^{2} + 10662 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
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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
−5.73812589338021216771892245360, −5.41627949932870405380881235136, −5.06489199016063625816894201352, −5.04850661913630374715683966257, −4.92241595564743753445556218881, −4.75649386291839883312593412153, −4.72864526896438551904392995787, −4.40838586209154441095090518196, −4.23848750455116035376804384910, −4.17876226032412850490795430184, −4.01466500307527323203232538551, −3.54531298097811069189650776905, −3.49951351285668897638779749280, −3.44574749391459005168799810958, −3.13452885624797282855874462960, −3.11865084233265932113536114836, −3.04524733287524019968809747837, −2.74383231450438970825911793480, −2.56109076747685554927755538732, −2.11771274823639781110416119215, −1.70978324994814382881067450747, −1.70850946274499909800647373114, −1.68299960504030342333200891621, −1.03754198720654088484727612719, −0.43139058921706822383338988331,
0.43139058921706822383338988331, 1.03754198720654088484727612719, 1.68299960504030342333200891621, 1.70850946274499909800647373114, 1.70978324994814382881067450747, 2.11771274823639781110416119215, 2.56109076747685554927755538732, 2.74383231450438970825911793480, 3.04524733287524019968809747837, 3.11865084233265932113536114836, 3.13452885624797282855874462960, 3.44574749391459005168799810958, 3.49951351285668897638779749280, 3.54531298097811069189650776905, 4.01466500307527323203232538551, 4.17876226032412850490795430184, 4.23848750455116035376804384910, 4.40838586209154441095090518196, 4.72864526896438551904392995787, 4.75649386291839883312593412153, 4.92241595564743753445556218881, 5.04850661913630374715683966257, 5.06489199016063625816894201352, 5.41627949932870405380881235136, 5.73812589338021216771892245360
Plot not available for L-functions of degree greater than 10.
