| L(s) = 1 | − 8·4-s + 24·9-s + 40·16-s − 192·36-s − 160·64-s + 324·81-s + 88·121-s + 127-s + 131-s + 137-s + 139-s + 960·144-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 + ⋯ |
| L(s) = 1 | − 4·4-s + 8·9-s + 10·16-s − 32·36-s − 20·64-s + 36·81-s + 8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 80·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.575703849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.575703849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} )^{4} \) |
| 5 | \( 1 - 48 T^{4} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - p T^{2} )^{8} \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 + 240 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 480 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 1440 T^{4} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 - p T^{2} )^{8} \) |
| 53 | \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 + 2640 T^{4} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( ( 1 + 10560 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - p T^{2} )^{8} \) |
| 83 | \( ( 1 - p T^{2} )^{8} \) |
| 89 | \( ( 1 - 12480 T^{4} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 18720 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.30455701381110122512836155141, −4.15550958319313131659879602478, −4.08274291790926662493067706496, −3.97363962383103706354742740207, −3.96640123125089604310358061588, −3.86169173375896259579069010958, −3.56548043117442713485720273588, −3.55509950221589065541971984332, −3.38096788044254400675961879516, −3.25845196593030679643956339127, −3.23192940031634471032156033311, −2.85913017237234035586513587731, −2.70597825221033768923506814197, −2.20813030549432332577024390540, −2.19493410253443957997810086737, −1.93866635695927036827263840947, −1.91863374377822642920842364650, −1.80814891417734309138307363278, −1.38740527189310000190276454998, −1.24222474089827617391807364323, −1.11177865793488242244567892667, −1.07604508229948423558510163813, −0.977384854848314537471369101836, −0.76958246808834459789797654455, −0.14639041080762727431872217361,
0.14639041080762727431872217361, 0.76958246808834459789797654455, 0.977384854848314537471369101836, 1.07604508229948423558510163813, 1.11177865793488242244567892667, 1.24222474089827617391807364323, 1.38740527189310000190276454998, 1.80814891417734309138307363278, 1.91863374377822642920842364650, 1.93866635695927036827263840947, 2.19493410253443957997810086737, 2.20813030549432332577024390540, 2.70597825221033768923506814197, 2.85913017237234035586513587731, 3.23192940031634471032156033311, 3.25845196593030679643956339127, 3.38096788044254400675961879516, 3.55509950221589065541971984332, 3.56548043117442713485720273588, 3.86169173375896259579069010958, 3.96640123125089604310358061588, 3.97363962383103706354742740207, 4.08274291790926662493067706496, 4.15550958319313131659879602478, 4.30455701381110122512836155141
Plot not available for L-functions of degree greater than 10.
