| L(s) = 1 | − 1.51e3·17-s + 868·25-s + 4.53e3·41-s + 3.33e3·49-s + 1.96e3·73-s − 3.78e4·89-s − 6.47e4·97-s − 5.59e4·113-s − 3.85e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.79e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 5.23·17-s + 1.38·25-s + 2.69·41-s + 1.38·49-s + 0.367·73-s − 4.77·89-s − 6.88·97-s − 4.38·113-s − 2.63·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 0.628·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7611711750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7611711750\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 434 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 34 p^{2} T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 19286 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8974 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 378 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 244118 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 513026 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1054706 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1796866 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 509618 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 1134 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 6027926 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 614786 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 14781362 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 24188822 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 27096818 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 25281514 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 30248066 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 490 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 70060162 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 26999126 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9450 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 16198 T + p^{4} T^{2} )^{4} \) |
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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
−7.00827658052646488458637027540, −6.86351281364805529025644287017, −6.71576673920487007334919496472, −6.40995056148928232877072268871, −6.36271355677910109649452071351, −5.85019395063592791889756792675, −5.71721913864899059362348294414, −5.28883432298893177481295287814, −5.20110275218598801678904805200, −4.83645651019798303766767125305, −4.54264284707536644667130227582, −4.20298571781493595524956392714, −4.03766873419552069030424299853, −4.00794489490895659197771187578, −3.92211209088697186174793692609, −2.86607387686265073371703449071, −2.65568790785818874463165444283, −2.65562558776526378870203105367, −2.60399164211473347214395909788, −2.08062851068365935109520138300, −1.58116688426639055142350405163, −1.35301556664270658431856033654, −1.03376630698073127002473065687, −0.30720929986868084488305151385, −0.19644041876716548101890396720,
0.19644041876716548101890396720, 0.30720929986868084488305151385, 1.03376630698073127002473065687, 1.35301556664270658431856033654, 1.58116688426639055142350405163, 2.08062851068365935109520138300, 2.60399164211473347214395909788, 2.65562558776526378870203105367, 2.65568790785818874463165444283, 2.86607387686265073371703449071, 3.92211209088697186174793692609, 4.00794489490895659197771187578, 4.03766873419552069030424299853, 4.20298571781493595524956392714, 4.54264284707536644667130227582, 4.83645651019798303766767125305, 5.20110275218598801678904805200, 5.28883432298893177481295287814, 5.71721913864899059362348294414, 5.85019395063592791889756792675, 6.36271355677910109649452071351, 6.40995056148928232877072268871, 6.71576673920487007334919496472, 6.86351281364805529025644287017, 7.00827658052646488458637027540
