| L(s) = 1 | − 32·16-s + 464·61-s + 158·81-s − 968·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 2·16-s + 7.60·61-s + 1.95·81-s − 8·121-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 + 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 + 0.00429·233-s + 0.00418·239-s + 0.00414·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.557648559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.557648559\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 3 | \( 1 - 158 T^{4} + p^{8} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 1922 T^{4} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 13 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 23 | \( ( 1 + 211202 T^{4} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 37 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 62 T + p^{2} T^{2} )^{4}( 1 + 62 T + p^{2} T^{2} )^{4} \) |
| 43 | \( ( 1 - 2519518 T^{4} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 9618242 T^{4} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 61 | \( ( 1 - 58 T + p^{2} T^{2} )^{8} \) |
| 67 | \( ( 1 - 20249758 T^{4} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 73 | \( ( 1 + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 83 | \( ( 1 - 30884638 T^{4} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 4322 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + p^{4} T^{4} )^{4} \) |
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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
−5.05658923567129911110545178892, −5.05297733741122225572267906921, −4.80021051534907457540038139210, −4.56505147808026510224091319751, −4.29783013880168640671825584926, −4.24084581457560996678754038079, −4.02685239622191132579588469194, −3.85515247717998688696349136625, −3.71704762842102883031747245170, −3.67386664827684195378902101700, −3.58916855024179296334451131519, −3.42286592142679672135981607499, −2.83309099426222078254464425378, −2.83266212644624136824461764023, −2.55630302627029258220329032202, −2.46100137552316105662496765749, −2.40819655387758273126825327719, −2.24695129792930421786052948045, −1.87003258489650891768893064234, −1.77730539435393950272866891508, −1.28269919399298145560803327589, −1.08010735404443029883024888508, −0.988093556199959430553669013559, −0.40542564938622657307734070176, −0.26485180946573081272478549024,
0.26485180946573081272478549024, 0.40542564938622657307734070176, 0.988093556199959430553669013559, 1.08010735404443029883024888508, 1.28269919399298145560803327589, 1.77730539435393950272866891508, 1.87003258489650891768893064234, 2.24695129792930421786052948045, 2.40819655387758273126825327719, 2.46100137552316105662496765749, 2.55630302627029258220329032202, 2.83266212644624136824461764023, 2.83309099426222078254464425378, 3.42286592142679672135981607499, 3.58916855024179296334451131519, 3.67386664827684195378902101700, 3.71704762842102883031747245170, 3.85515247717998688696349136625, 4.02685239622191132579588469194, 4.24084581457560996678754038079, 4.29783013880168640671825584926, 4.56505147808026510224091319751, 4.80021051534907457540038139210, 5.05297733741122225572267906921, 5.05658923567129911110545178892
Plot not available for L-functions of degree greater than 10.
