| L(s) = 1 | − 7·9-s + 28·19-s − 168·31-s + 196·49-s − 32·61-s + 48·79-s − 32·81-s + 352·109-s − 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s − 196·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 7/9·9-s + 1.47·19-s − 5.41·31-s + 4·49-s − 0.524·61-s + 0.607·79-s − 0.395·81-s + 3.22·109-s − 0.545·121-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 + 2.81·169-s − 1.14·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4987015220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4987015220\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 7 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 p T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 567 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 662 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 582 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 1138 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3087 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2262 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3462 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2562 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 6953 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8982 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9433 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8927 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )^{2}( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 13918 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85726128012539186852687018220, −6.64263594787134029475049971082, −6.27659817678836439021130886047, −5.86343174535615408539281897524, −5.74809075835268551589423318747, −5.73151238297019246861626465803, −5.42198162734690999409322330562, −5.40598260020880151674722326711, −5.07402018450261484579416632750, −4.77471490639852831750728324547, −4.39608834725759111131032366875, −4.19833468788243842648382983305, −4.04433729627616871594220781514, −3.57234188002098584414905984081, −3.35871393476091376541860979664, −3.27548720131382942772993140707, −3.27032738525696349516933458369, −2.40962405508497959720157168687, −2.35648835150368943190261333084, −2.17630845650918623954691431112, −1.84882816263339239707993151531, −1.26231668035046381412419823359, −1.16066254331973412799473948018, −0.61165414239177065140638244252, −0.11540378507972748670793894992,
0.11540378507972748670793894992, 0.61165414239177065140638244252, 1.16066254331973412799473948018, 1.26231668035046381412419823359, 1.84882816263339239707993151531, 2.17630845650918623954691431112, 2.35648835150368943190261333084, 2.40962405508497959720157168687, 3.27032738525696349516933458369, 3.27548720131382942772993140707, 3.35871393476091376541860979664, 3.57234188002098584414905984081, 4.04433729627616871594220781514, 4.19833468788243842648382983305, 4.39608834725759111131032366875, 4.77471490639852831750728324547, 5.07402018450261484579416632750, 5.40598260020880151674722326711, 5.42198162734690999409322330562, 5.73151238297019246861626465803, 5.74809075835268551589423318747, 5.86343174535615408539281897524, 6.27659817678836439021130886047, 6.64263594787134029475049971082, 6.85726128012539186852687018220
