| L(s) = 1 | − 64·19-s + 16·31-s − 32·49-s − 80·61-s − 400·79-s − 208·109-s + 176·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 976·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 3.36·19-s + 0.516·31-s − 0.653·49-s − 1.31·61-s − 5.06·79-s − 1.90·109-s + 1.45·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.77·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1120776467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1120776467\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 16 T^{2} - 894 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 488 T^{2} + 115218 T^{4} - 488 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 1612 T^{2} + 1157478 T^{4} + 1612 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 1566 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1544 T^{2} + 598866 T^{4} - 1544 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 5876 T^{2} + 14893446 T^{4} - 5876 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 340 T^{2} - 372378 T^{4} + 340 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 1876 T^{2} - 4075194 T^{4} + 1876 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 20 T + 4302 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 3202 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 11948 T^{2} + 73770918 T^{4} - 11948 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 100 T + 11742 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 21652 T^{2} + 211289478 T^{4} + 21652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 14444 T^{2} + 213036006 T^{4} - 14444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.16059750009955888314079359905, −4.12186670954146915772302108931, −3.96686182859344493301768107409, −3.92828802391607416482579294081, −3.58668376616940826839736702601, −3.28661728117934919873632028502, −3.24977788739380644352820547596, −3.23554673231470119031588521265, −2.95126914759367068566971553982, −2.83086382875173089209522586443, −2.82002102925084434170671162410, −2.59495869733104063950099638473, −2.47711546948951845089851198538, −2.22957382444725797769773563285, −1.98501816334522092710717941286, −1.95243950326073228082822925611, −1.86789613502046201847544898574, −1.58582430522006536091612647738, −1.47004919948285095665611445927, −1.20591895565844682203200462623, −1.18216442679400309577407966697, −0.69752858211954094291631151459, −0.56698035555902834904656178301, −0.27561749974600317486634808155, −0.04088042883160799151213613924,
0.04088042883160799151213613924, 0.27561749974600317486634808155, 0.56698035555902834904656178301, 0.69752858211954094291631151459, 1.18216442679400309577407966697, 1.20591895565844682203200462623, 1.47004919948285095665611445927, 1.58582430522006536091612647738, 1.86789613502046201847544898574, 1.95243950326073228082822925611, 1.98501816334522092710717941286, 2.22957382444725797769773563285, 2.47711546948951845089851198538, 2.59495869733104063950099638473, 2.82002102925084434170671162410, 2.83086382875173089209522586443, 2.95126914759367068566971553982, 3.23554673231470119031588521265, 3.24977788739380644352820547596, 3.28661728117934919873632028502, 3.58668376616940826839736702601, 3.92828802391607416482579294081, 3.96686182859344493301768107409, 4.12186670954146915772302108931, 4.16059750009955888314079359905
Plot not available for L-functions of degree greater than 10.
