| L(s) = 1 | − 4-s − 8·7-s − 8·13-s − 6·16-s − 14·19-s − 13·25-s + 8·28-s − 14·31-s + 16·37-s − 16·43-s + 22·49-s + 8·52-s − 14·61-s + 9·64-s + 2·67-s − 8·73-s + 14·76-s − 20·79-s + 64·91-s − 2·97-s + 13·100-s − 8·103-s − 28·109-s + 48·112-s + 14·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 3.02·7-s − 2.21·13-s − 3/2·16-s − 3.21·19-s − 2.59·25-s + 1.51·28-s − 2.51·31-s + 2.63·37-s − 2.43·43-s + 22/7·49-s + 1.10·52-s − 1.79·61-s + 9/8·64-s + 0.244·67-s − 0.936·73-s + 1.60·76-s − 2.25·79-s + 6.70·91-s − 0.203·97-s + 1.29·100-s − 0.788·103-s − 2.68·109-s + 4.53·112-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + T^{2} + 7 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 13 T^{2} + 91 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 5 T^{2} + 583 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 753 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 64 T^{2} + 2206 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 7 T + 73 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 8 T + 85 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 124 T^{2} + 7081 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 175 T^{2} + 12043 T^{4} + 175 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 127 T^{2} + 8119 T^{4} + 127 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 109 T^{2} + 9871 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 7 T + 73 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - T + 123 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 109 T^{2} + 12271 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 10 T + 163 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 244 T^{2} + 28057 T^{4} + 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 4 T^{2} + 5721 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.40674683574967194636288663023, −7.14923710979333734322951701235, −7.08011402581972262837050978198, −6.78557986644053019388249232047, −6.59331139707329734387532911045, −6.34259358771959291204506303583, −6.18931431894214012683287509878, −6.14456172844851917404305537478, −5.79811605319430001964764881271, −5.64721491664405964750415749357, −5.03951266837666531460868816361, −5.02113543791700403136738876090, −4.87830124815686636564628323552, −4.37679416259494002266349194527, −4.11785191893443414325657861941, −4.10481830365529033694937838549, −3.87641489913800373431100752844, −3.46033612761326046025263372812, −3.37411237312926667952477792284, −2.69236487425435643261613352620, −2.63464502172698237464699743788, −2.52935291967428240028390260955, −2.21209802837975587142245198269, −1.66859312033851710556084680056, −1.55206247761221413963865314443, 0, 0, 0, 0,
1.55206247761221413963865314443, 1.66859312033851710556084680056, 2.21209802837975587142245198269, 2.52935291967428240028390260955, 2.63464502172698237464699743788, 2.69236487425435643261613352620, 3.37411237312926667952477792284, 3.46033612761326046025263372812, 3.87641489913800373431100752844, 4.10481830365529033694937838549, 4.11785191893443414325657861941, 4.37679416259494002266349194527, 4.87830124815686636564628323552, 5.02113543791700403136738876090, 5.03951266837666531460868816361, 5.64721491664405964750415749357, 5.79811605319430001964764881271, 6.14456172844851917404305537478, 6.18931431894214012683287509878, 6.34259358771959291204506303583, 6.59331139707329734387532911045, 6.78557986644053019388249232047, 7.08011402581972262837050978198, 7.14923710979333734322951701235, 7.40674683574967194636288663023
