| L(s) = 1 | − 40·25-s − 8·49-s − 80·73-s − 112·97-s + 136·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-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 + ⋯ |
| L(s) = 1 | − 8·25-s − 8/7·49-s − 9.36·73-s − 11.3·97-s + 12.3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.578069941\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.578069941\times10^{-5}\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{8} \) |
| good | 5 | \( ( 1 + 2 p T^{2} + 54 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{4} \) |
| 11 | \( ( 1 - 34 T^{2} + 510 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 - 32 T^{2} + 510 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 22 T^{2} + 174 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{8} \) |
| 23 | \( ( 1 + 70 T^{2} + 2262 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 8 T^{2} + 354 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{8} \) |
| 37 | \( ( 1 - 80 T^{2} + 3582 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 26 T^{2} + 3342 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 + 20 T^{2} - 1578 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{8} \) |
| 53 | \( ( 1 + 136 T^{2} + 8898 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 - p T^{2} )^{16} \) |
| 61 | \( ( 1 - 14 T + p T^{2} )^{8}( 1 + 14 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 200 T^{2} + 18222 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 254 T^{2} + 26022 T^{4} + 254 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 + 80 T^{2} + 7278 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 4 T^{2} + 5382 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 166 T^{2} + 22542 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.99713703496434975495584851528, −1.98020939971037556012887490198, −1.94838550390134719203087879337, −1.85634811614904717288564751826, −1.77561053162430344545300296820, −1.56162107453976556137600948323, −1.54544819595916813189407937594, −1.52320737432118670071017131966, −1.51827853615113300400126193082, −1.46922200587654134678064935254, −1.42702576049018909315258581933, −1.30953129224655850211064709490, −1.25213980044446213158122507865, −1.11144066092977623967537942955, −1.06934599763782384526145005732, −0.990617429914980236547881348023, −0.880122482303472240044464635947, −0.816363954498810292737362315756, −0.58883657103295071608592768301, −0.42525949503236434345250898525, −0.41740603356420236361077799977, −0.24043438150772755665033707084, −0.19018907050634605734134476642, −0.15906754049462487837373373401, −0.00115759301572620047819726424,
0.00115759301572620047819726424, 0.15906754049462487837373373401, 0.19018907050634605734134476642, 0.24043438150772755665033707084, 0.41740603356420236361077799977, 0.42525949503236434345250898525, 0.58883657103295071608592768301, 0.816363954498810292737362315756, 0.880122482303472240044464635947, 0.990617429914980236547881348023, 1.06934599763782384526145005732, 1.11144066092977623967537942955, 1.25213980044446213158122507865, 1.30953129224655850211064709490, 1.42702576049018909315258581933, 1.46922200587654134678064935254, 1.51827853615113300400126193082, 1.52320737432118670071017131966, 1.54544819595916813189407937594, 1.56162107453976556137600948323, 1.77561053162430344545300296820, 1.85634811614904717288564751826, 1.94838550390134719203087879337, 1.98020939971037556012887490198, 1.99713703496434975495584851528
Plot not available for L-functions of degree greater than 10.
