| L(s) = 1 | − 4·3-s + 8·9-s − 8·11-s + 8·13-s + 16·23-s + 8·25-s − 12·27-s + 32·33-s + 8·37-s − 32·39-s + 16·47-s − 2·49-s + 8·59-s + 40·61-s − 64·69-s + 24·71-s − 24·73-s − 32·75-s + 23·81-s − 40·83-s − 24·97-s − 64·99-s − 24·107-s + 24·109-s − 32·111-s + 64·117-s − 4·121-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 8/3·9-s − 2.41·11-s + 2.21·13-s + 3.33·23-s + 8/5·25-s − 2.30·27-s + 5.57·33-s + 1.31·37-s − 5.12·39-s + 2.33·47-s − 2/7·49-s + 1.04·59-s + 5.12·61-s − 7.70·69-s + 2.84·71-s − 2.80·73-s − 3.69·75-s + 23/9·81-s − 4.39·83-s − 2.43·97-s − 6.43·99-s − 2.32·107-s + 2.29·109-s − 3.03·111-s + 5.91·117-s − 0.363·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.355208714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355208714\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 690 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3670 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 2646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 104 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 6934 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 20 T + 264 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.41843034126456560991944332068, −7.08118385336549422324909323453, −6.96071656128207588959829549560, −6.92847772272591650269064830278, −6.69862903310707563641387628365, −6.30727769041174894730966100778, −6.05555522058257940805077250026, −5.82802787239449586360302673728, −5.60962065453218825339127858738, −5.29865505945942911404291085520, −5.20407485283440960723994260811, −5.10664896202658723678167964576, −5.01339835540800440140581583751, −4.41497978454695720328962575441, −4.02368219379534441020657121560, −3.95668556758420235371392159104, −3.79791086170925282233638470151, −2.98018689520208026215155644446, −2.88018013703640576032443937339, −2.68225124675847116219726995264, −2.44043697319980317257542342853, −1.57797449405011527201244697217, −1.22787886560167106695665827260, −0.69783016129076199385879112074, −0.67680233468151085231554848849,
0.67680233468151085231554848849, 0.69783016129076199385879112074, 1.22787886560167106695665827260, 1.57797449405011527201244697217, 2.44043697319980317257542342853, 2.68225124675847116219726995264, 2.88018013703640576032443937339, 2.98018689520208026215155644446, 3.79791086170925282233638470151, 3.95668556758420235371392159104, 4.02368219379534441020657121560, 4.41497978454695720328962575441, 5.01339835540800440140581583751, 5.10664896202658723678167964576, 5.20407485283440960723994260811, 5.29865505945942911404291085520, 5.60962065453218825339127858738, 5.82802787239449586360302673728, 6.05555522058257940805077250026, 6.30727769041174894730966100778, 6.69862903310707563641387628365, 6.92847772272591650269064830278, 6.96071656128207588959829549560, 7.08118385336549422324909323453, 7.41843034126456560991944332068
