| L(s) = 1 | − 2·2-s + 3·4-s − 8·8-s + 9·16-s − 4·25-s + 16·29-s − 6·32-s + 16·37-s + 18·49-s + 8·50-s − 16·53-s − 32·58-s + 11·64-s − 32·74-s − 36·98-s − 12·100-s + 32·106-s − 24·109-s − 16·113-s + 48·116-s + 32·121-s + 127-s + 12·128-s + 131-s + 137-s + 139-s + 48·148-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 2.82·8-s + 9/4·16-s − 4/5·25-s + 2.97·29-s − 1.06·32-s + 2.63·37-s + 18/7·49-s + 1.13·50-s − 2.19·53-s − 4.20·58-s + 11/8·64-s − 3.71·74-s − 3.63·98-s − 6/5·100-s + 3.10·106-s − 2.29·109-s − 1.50·113-s + 4.45·116-s + 2.90·121-s + 0.0887·127-s + 1.06·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.94·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.536026180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536026180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 - 18 T^{2} + 162 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 - 16 T^{2} + 238 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 48 T^{2} + 1230 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 18 T^{2} + 1122 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 68 T^{2} + 2806 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 114 T^{2} + 6114 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 62 T^{2} + 4002 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 112 T^{2} + 6766 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 50 T^{2} + 610 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 192 T^{2} + 19230 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 296 T^{2} + 34318 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 322 T^{2} + 39682 T^{4} + 322 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 336 T^{2} + 46430 T^{4} - 336 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.34397979057363308758542816005, −3.89296738918905296775091725735, −3.74906302593296204103428076947, −3.70827344333859387370691369277, −3.50595772621327673447257542001, −3.40241822691478255131656860466, −3.32201745632153607625969098917, −3.14405562969957160935382376387, −3.07325064906547227328779204667, −2.75064271701704446472580207342, −2.72774182159020843442189523624, −2.66752042686840965007719339485, −2.54848580843282777593179862978, −2.39810878495116680827079662790, −2.08686119532292454136582587625, −2.06005231889101507227453530606, −2.05583407559359198429732609538, −1.63894497498434354511403627438, −1.43768288994629461067865707136, −1.42941991105304954175810274080, −0.981807626920474700063787404827, −0.875926593191601924061454956438, −0.61115613827000443558099971085, −0.57735473070681414903071578754, −0.28827452098549073983828908392,
0.28827452098549073983828908392, 0.57735473070681414903071578754, 0.61115613827000443558099971085, 0.875926593191601924061454956438, 0.981807626920474700063787404827, 1.42941991105304954175810274080, 1.43768288994629461067865707136, 1.63894497498434354511403627438, 2.05583407559359198429732609538, 2.06005231889101507227453530606, 2.08686119532292454136582587625, 2.39810878495116680827079662790, 2.54848580843282777593179862978, 2.66752042686840965007719339485, 2.72774182159020843442189523624, 2.75064271701704446472580207342, 3.07325064906547227328779204667, 3.14405562969957160935382376387, 3.32201745632153607625969098917, 3.40241822691478255131656860466, 3.50595772621327673447257542001, 3.70827344333859387370691369277, 3.74906302593296204103428076947, 3.89296738918905296775091725735, 4.34397979057363308758542816005
Plot not available for L-functions of degree greater than 10.
