| L(s) = 1 | − 4·4-s + 4·16-s + 32·25-s − 64·37-s + 416·43-s + 16·64-s + 208·67-s − 416·79-s − 128·100-s + 248·109-s + 160·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 760·169-s − 1.66e3·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 1/4·16-s + 1.27·25-s − 1.72·37-s + 9.67·43-s + 1/4·64-s + 3.10·67-s − 5.26·79-s − 1.27·100-s + 2.27·109-s + 1.32·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4.49·169-s − 9.67·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1592798825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1592798825\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 16 T^{2} - 369 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 80 T^{2} - 8241 T^{4} - 80 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 512 T^{2} + 178623 T^{4} + 512 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 590 T^{2} + 217779 T^{4} + 590 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 608 T^{2} + 89823 T^{4} - 608 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - p T + p^{2} T^{2} )^{4}( 1 + p T + p^{2} T^{2} )^{4} \) |
| 37 | \( ( 1 + 16 T - 1113 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 128 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 52 T + p^{2} T^{2} )^{8} \) |
| 47 | \( ( 1 + 3362 T^{2} + 6423363 T^{4} + 3362 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 5330 T^{2} + 20518419 T^{4} - 5330 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 5906 T^{2} + 22763475 T^{4} + 5906 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6914 T^{2} + 33957555 T^{4} + 6914 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 52 T - 1785 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 + 2144 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 8546 T^{2} + 44635875 T^{4} + 8546 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 104 T + 4575 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 34 T + p^{2} T^{2} )^{4}( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 89 | \( ( 1 + 10496 T^{2} + 47423775 T^{4} + 10496 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 10370 T^{2} + p^{4} T^{4} )^{4} \) |
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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
−4.08897508553099291669857340829, −4.05728323014081302355498492663, −4.00173630717864800740184014161, −3.87475106111704264679430078517, −3.76159348810727315268540997405, −3.45541767891553326525560274121, −3.14195992459433403532709786234, −3.11549585126076218429334722240, −3.11108427663612944949034455380, −3.01899763950392449172835795302, −2.61258257681649529932823204328, −2.60386918675537119048704712433, −2.42308916534264778618435192354, −2.37103737867408959900075742429, −2.22106233778886646765961350702, −1.88196913434763835167470646822, −1.86701937019741116816137993847, −1.62983444320969843995399415032, −1.19793086511203118455852807188, −1.11839224021079960517888589040, −0.955334106245506702167768762875, −0.898555640256923677376455525854, −0.62455689424978458740766306723, −0.46692219008134979802435586626, −0.03079352378980187538175791333,
0.03079352378980187538175791333, 0.46692219008134979802435586626, 0.62455689424978458740766306723, 0.898555640256923677376455525854, 0.955334106245506702167768762875, 1.11839224021079960517888589040, 1.19793086511203118455852807188, 1.62983444320969843995399415032, 1.86701937019741116816137993847, 1.88196913434763835167470646822, 2.22106233778886646765961350702, 2.37103737867408959900075742429, 2.42308916534264778618435192354, 2.60386918675537119048704712433, 2.61258257681649529932823204328, 3.01899763950392449172835795302, 3.11108427663612944949034455380, 3.11549585126076218429334722240, 3.14195992459433403532709786234, 3.45541767891553326525560274121, 3.76159348810727315268540997405, 3.87475106111704264679430078517, 4.00173630717864800740184014161, 4.05728323014081302355498492663, 4.08897508553099291669857340829
Plot not available for L-functions of degree greater than 10.
