| L(s) = 1 | + 17·9-s + 44·19-s + 184·31-s + 68·49-s − 64·61-s + 272·79-s + 208·81-s − 704·109-s − 146·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 668·169-s + 748·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 17/9·9-s + 2.31·19-s + 5.93·31-s + 1.38·49-s − 1.04·61-s + 3.44·79-s + 2.56·81-s − 6.45·109-s − 1.20·121-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 + 3.95·169-s + 4.37·171-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.954092761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.954092761\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2}( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 263 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 202 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 422 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2482 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 527 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3158 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 4358 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 1922 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 3791 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1258 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 457 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 13463 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 13007 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18334 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.67436252943759234519554843560, −6.51280482539555639546734414919, −6.38188353578158092494323675043, −6.24281035635325545081514209020, −6.06086552444017296486121887015, −5.56633841750845384883456934740, −5.31793937448084755863499573908, −5.02368519325453347194959849554, −4.96733628606506432308480739509, −4.90239361223595425291033578893, −4.43437440465103462244890480756, −4.15915230797381405777562245236, −4.02759186349882828533524722298, −3.96113236563399997059489270789, −3.40341900252394626303738754827, −3.17895047070393791056631547032, −3.00394983330609464481894035068, −2.46404992187928207551284584523, −2.43601108656783332642764654077, −2.33134638063580327888358275835, −1.43245983898348329395015517444, −1.22430071601473021767067985341, −1.13954497021868205143885352021, −0.953048063067222380272876681557, −0.36663172597538225547862270408,
0.36663172597538225547862270408, 0.953048063067222380272876681557, 1.13954497021868205143885352021, 1.22430071601473021767067985341, 1.43245983898348329395015517444, 2.33134638063580327888358275835, 2.43601108656783332642764654077, 2.46404992187928207551284584523, 3.00394983330609464481894035068, 3.17895047070393791056631547032, 3.40341900252394626303738754827, 3.96113236563399997059489270789, 4.02759186349882828533524722298, 4.15915230797381405777562245236, 4.43437440465103462244890480756, 4.90239361223595425291033578893, 4.96733628606506432308480739509, 5.02368519325453347194959849554, 5.31793937448084755863499573908, 5.56633841750845384883456934740, 6.06086552444017296486121887015, 6.24281035635325545081514209020, 6.38188353578158092494323675043, 6.51280482539555639546734414919, 6.67436252943759234519554843560
