| L(s) = 1 | − 8·13-s − 40·19-s + 68·25-s + 40·31-s + 32·37-s − 16·43-s + 14·49-s − 272·61-s + 168·67-s + 144·73-s + 232·79-s + 96·97-s − 328·103-s + 240·109-s + 368·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 412·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 0.615·13-s − 2.10·19-s + 2.71·25-s + 1.29·31-s + 0.864·37-s − 0.372·43-s + 2/7·49-s − 4.45·61-s + 2.50·67-s + 1.97·73-s + 2.93·79-s + 0.989·97-s − 3.18·103-s + 2.20·109-s + 3.04·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.43·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.545929477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545929477\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2294 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 368 T^{2} + 62690 T^{4} - 368 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 230 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 452 T^{2} + 215318 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20 T + 30 p T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1808 T^{2} + 1354946 T^{4} - 1808 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 112 T^{2} - 475102 T^{4} + 112 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 20 T + 1994 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 16 T + 1794 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6436 T^{2} + 15997974 T^{4} - 6436 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 2706 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2644 T^{2} + 6281574 T^{4} - 2644 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 3440 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6836 T^{2} + 27136646 T^{4} - 6836 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 136 T + 12038 T^{2} + 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 84 T + 5254 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4496 T^{2} + 27200834 T^{4} - 4496 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 72 T + 11926 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 116 T + 8678 T^{2} - 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8228 T^{2} + 33565286 T^{4} - 8228 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 17852 T^{2} + 192738390 T^{4} + 17852 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 48 T + 11302 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.674397162593379581250761020188, −8.182379977098048263517903785445, −8.011625873238990525090667926650, −7.903112900082728470400856156350, −7.62819832588394683045380211344, −7.00785676547991377380077892694, −7.00148450453633760505673440242, −6.60285736531223964650755129485, −6.38171383400624830370502501788, −6.36985342416908602363016394899, −5.93261422167387465727958254359, −5.45655766171678751392568251182, −5.19359359860101995145910308877, −4.86534100919532718250294012419, −4.62780881662315258591889107085, −4.36675009357007585314555427180, −4.18174897015607115737656303209, −3.47931478857677708526053438035, −3.33060934678352650245178698008, −2.85799996109175556790418419789, −2.48510880231824571049694800520, −2.18839727466552788645353100852, −1.66974672275786460762823963245, −0.941662771887434550174381277187, −0.49277711635411603065246499232,
0.49277711635411603065246499232, 0.941662771887434550174381277187, 1.66974672275786460762823963245, 2.18839727466552788645353100852, 2.48510880231824571049694800520, 2.85799996109175556790418419789, 3.33060934678352650245178698008, 3.47931478857677708526053438035, 4.18174897015607115737656303209, 4.36675009357007585314555427180, 4.62780881662315258591889107085, 4.86534100919532718250294012419, 5.19359359860101995145910308877, 5.45655766171678751392568251182, 5.93261422167387465727958254359, 6.36985342416908602363016394899, 6.38171383400624830370502501788, 6.60285736531223964650755129485, 7.00148450453633760505673440242, 7.00785676547991377380077892694, 7.62819832588394683045380211344, 7.903112900082728470400856156350, 8.011625873238990525090667926650, 8.182379977098048263517903785445, 8.674397162593379581250761020188
