| L(s) = 1 | − 2·2-s − 3·4-s + 10·8-s − 4·11-s + 8·22-s + 20·23-s − 14·25-s + 4·29-s − 18·32-s − 16·37-s − 12·43-s + 12·44-s − 40·46-s + 28·50-s − 16·53-s − 8·58-s + 11·64-s − 12·67-s − 4·71-s + 32·74-s − 40·79-s + 24·86-s − 40·88-s − 60·92-s + 42·100-s + 32·106-s − 12·107-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3/2·4-s + 3.53·8-s − 1.20·11-s + 1.70·22-s + 4.17·23-s − 2.79·25-s + 0.742·29-s − 3.18·32-s − 2.63·37-s − 1.82·43-s + 1.80·44-s − 5.89·46-s + 3.95·50-s − 2.19·53-s − 1.05·58-s + 11/8·64-s − 1.46·67-s − 0.474·71-s + 3.71·74-s − 4.50·79-s + 2.58·86-s − 4.26·88-s − 6.25·92-s + 21/5·100-s + 3.10·106-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( ( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 14 T^{2} + 94 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 454 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 30 T^{2} + 902 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 88 T^{2} + 3538 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 45 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 30 T^{2} + 2462 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 6 T + 15 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 22 T^{2} - 906 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 77 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 38 p T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 5262 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 123 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 123 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 8182 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 253 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 206 T^{2} + 24262 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 262 T^{2} + 35374 T^{4} + 262 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.49888412391409346109967251294, −6.06374592074659581881612591199, −5.94309058596093700632412785862, −5.66927508676049119015465663751, −5.62395318018456381666138855926, −5.11046463393104704101241153180, −5.07805143340507795157227559589, −5.02092125370246536587608499055, −4.97257222144476557887819455931, −4.63007547215530624146475634314, −4.41322336534226989524796469781, −4.27199186284697979777725902471, −3.97048683728751849733674734290, −3.79834575589675819372149389674, −3.37440860574540312836679384485, −3.31133431721467592308520451539, −3.19983593181019503833208028765, −2.81811625199466858282188377762, −2.66455995315241967121597814553, −2.19108860104994771447599574560, −2.18953487199178929929037154603, −1.49319079467808264052890179337, −1.24791344974786545368513351054, −1.22088366582123841729526059483, −1.15461416223493277683239009570, 0, 0, 0, 0,
1.15461416223493277683239009570, 1.22088366582123841729526059483, 1.24791344974786545368513351054, 1.49319079467808264052890179337, 2.18953487199178929929037154603, 2.19108860104994771447599574560, 2.66455995315241967121597814553, 2.81811625199466858282188377762, 3.19983593181019503833208028765, 3.31133431721467592308520451539, 3.37440860574540312836679384485, 3.79834575589675819372149389674, 3.97048683728751849733674734290, 4.27199186284697979777725902471, 4.41322336534226989524796469781, 4.63007547215530624146475634314, 4.97257222144476557887819455931, 5.02092125370246536587608499055, 5.07805143340507795157227559589, 5.11046463393104704101241153180, 5.62395318018456381666138855926, 5.66927508676049119015465663751, 5.94309058596093700632412785862, 6.06374592074659581881612591199, 6.49888412391409346109967251294
