| L(s) = 1 | − 2·2-s + 4-s − 4·7-s + 2·8-s + 6·13-s + 8·14-s − 4·16-s + 12·17-s + 16·19-s − 8·25-s − 12·26-s − 4·28-s + 2·32-s − 24·34-s − 32·38-s + 2·43-s − 12·47-s − 6·49-s + 16·50-s + 6·52-s − 8·56-s + 24·59-s + 2·61-s + 3·64-s − 6·67-s + 12·68-s + 12·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.51·7-s + 0.707·8-s + 1.66·13-s + 2.13·14-s − 16-s + 2.91·17-s + 3.67·19-s − 8/5·25-s − 2.35·26-s − 0.755·28-s + 0.353·32-s − 4.11·34-s − 5.19·38-s + 0.304·43-s − 1.75·47-s − 6/7·49-s + 2.26·50-s + 0.832·52-s − 1.06·56-s + 3.12·59-s + 0.256·61-s + 3/8·64-s − 0.733·67-s + 1.45·68-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9355029660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9355029660\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2463 T^{4} - 528 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 59 T^{2} + 46 T^{3} + 1948 T^{4} + 46 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 152 T^{2} + 1248 T^{3} + 10863 T^{4} + 1248 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 320 T^{2} - 3312 T^{3} + 28071 T^{4} - 3312 p T^{5} + 320 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 390 T^{3} + 540 T^{4} + 390 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 18 T + 275 T^{2} - 3006 T^{3} + 30180 T^{4} - 3006 p T^{5} + 275 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.332475608197135665206761189290, −8.126480324678945877205973790325, −7.84634785043436200128682521670, −7.82959039493287007573578757840, −7.57372737806915186569548808838, −7.05981310357090466873499900579, −6.91437204981538512053671243721, −6.86710933531952034498400281132, −6.23212832585137339153002132835, −6.19045815224334056617204940269, −5.73281301284204617362476646098, −5.46639436710852106939844156242, −5.38613130679508868604721969862, −5.17520897728168386702038101616, −4.73731963071845419337759652148, −4.11243307095718964979369491167, −3.70359059311434464693172651942, −3.58987002371699899314197121743, −3.41142410324086143876021406839, −3.11798061047862798436001259007, −2.75871834402987645767145305650, −2.03417745204714624361117094823, −1.35930594811035093376430293133, −1.17122575279107761539823766299, −0.67380810995184899578608453205,
0.67380810995184899578608453205, 1.17122575279107761539823766299, 1.35930594811035093376430293133, 2.03417745204714624361117094823, 2.75871834402987645767145305650, 3.11798061047862798436001259007, 3.41142410324086143876021406839, 3.58987002371699899314197121743, 3.70359059311434464693172651942, 4.11243307095718964979369491167, 4.73731963071845419337759652148, 5.17520897728168386702038101616, 5.38613130679508868604721969862, 5.46639436710852106939844156242, 5.73281301284204617362476646098, 6.19045815224334056617204940269, 6.23212832585137339153002132835, 6.86710933531952034498400281132, 6.91437204981538512053671243721, 7.05981310357090466873499900579, 7.57372737806915186569548808838, 7.82959039493287007573578757840, 7.84634785043436200128682521670, 8.126480324678945877205973790325, 8.332475608197135665206761189290
