| L(s) = 1 | − 48·9-s − 192·41-s − 64·49-s + 1.11e3·81-s − 1.20e3·89-s − 664·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 184·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 5.33·9-s − 4.68·41-s − 1.30·49-s + 13.7·81-s − 13.4·89-s − 5.48·121-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 − 1.08·169-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 + 0.00436·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05701543119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05701543119\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 + 4 p T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 + 16 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 166 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 122 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 2 p T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 944 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 - 466 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 1906 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2522 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 24 T + p^{2} T^{2} )^{8} \) |
| 43 | \( ( 1 + 524 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 4304 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 - 94 T + p^{2} T^{2} )^{4}( 1 + 94 T + p^{2} T^{2} )^{4} \) |
| 59 | \( ( 1 + 5062 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 6758 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 8924 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 742 T^{2} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 13484 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 + 150 T + p^{2} T^{2} )^{8} \) |
| 97 | \( ( 1 + 18362 T^{2} + p^{4} T^{4} )^{4} \) |
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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
−3.57048067740562284419891333899, −3.56668867896424625061778278158, −3.46601142154520936953526557783, −3.39983450211778955655705510787, −3.22368375981190681791227219888, −3.10031579770823346985118806275, −2.87685563284689852477634999461, −2.80704749110649537708242768532, −2.76070788420058900478803641868, −2.59367254281758105588067895405, −2.57059714518909647948438369759, −2.50311456572889339952874404994, −2.43185556759592871571947130949, −2.03326581318882080026837706999, −1.73544922519266752793603230483, −1.72795763786997972156071305715, −1.59499274385542949831111121721, −1.42710503075306104873464328668, −1.31065858653785764867921759761, −1.11913119654462886924092657661, −0.910318445250676894435375568478, −0.42423836973280665898431185451, −0.27972146391001892026107127065, −0.12345814557368247087006107480, −0.11181123133787374282357976657,
0.11181123133787374282357976657, 0.12345814557368247087006107480, 0.27972146391001892026107127065, 0.42423836973280665898431185451, 0.910318445250676894435375568478, 1.11913119654462886924092657661, 1.31065858653785764867921759761, 1.42710503075306104873464328668, 1.59499274385542949831111121721, 1.72795763786997972156071305715, 1.73544922519266752793603230483, 2.03326581318882080026837706999, 2.43185556759592871571947130949, 2.50311456572889339952874404994, 2.57059714518909647948438369759, 2.59367254281758105588067895405, 2.76070788420058900478803641868, 2.80704749110649537708242768532, 2.87685563284689852477634999461, 3.10031579770823346985118806275, 3.22368375981190681791227219888, 3.39983450211778955655705510787, 3.46601142154520936953526557783, 3.56668867896424625061778278158, 3.57048067740562284419891333899
Plot not available for L-functions of degree greater than 10.
