| L(s) = 1 | − 4·4-s − 12·11-s + 7·16-s + 4·19-s − 12·29-s − 4·31-s − 24·41-s + 48·44-s − 16·49-s − 24·59-s + 8·61-s − 8·64-s − 36·71-s − 16·76-s + 16·79-s − 12·89-s − 24·101-s + 16·109-s + 48·116-s + 52·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3.61·11-s + 7/4·16-s + 0.917·19-s − 2.22·29-s − 0.718·31-s − 3.74·41-s + 7.23·44-s − 2.28·49-s − 3.12·59-s + 1.02·61-s − 64-s − 4.27·71-s − 1.83·76-s + 1.80·79-s − 1.27·89-s − 2.38·101-s + 1.53·109-s + 4.45·116-s + 4.72·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 135 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 786 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 88 T^{2} + 2991 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 88 T^{2} + 3906 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 10791 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 196 T^{2} + 15174 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 184 T^{2} + 17199 T^{4} + 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 237 p T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 15606 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.08132901188523211428703620602, −6.71902792154083534705674743420, −6.31759477770257476063171847735, −6.25199012258728612563505704159, −6.07187244337707152199807554226, −5.63357519933775621292865858143, −5.55788632991090118603725359419, −5.35697581715995615644580918662, −5.21559152981152156899512728043, −4.85832966913033718092105022409, −4.84872088774248627400816145810, −4.76365493224118229097579702390, −4.66611721763567886754645592883, −4.05103705666245124434027104217, −3.77630586526435975535751320334, −3.65230448601682388537914482538, −3.59945666760147367008141919173, −3.00828099489773107372633597923, −2.99256636265211871951421339073, −2.81452384711716862351284369338, −2.39956928953848527145903951004, −2.14295196954076398741079532985, −1.67123418952172474069803198454, −1.36335420204119645856285988328, −1.30389575925123182934085830465, 0, 0, 0, 0,
1.30389575925123182934085830465, 1.36335420204119645856285988328, 1.67123418952172474069803198454, 2.14295196954076398741079532985, 2.39956928953848527145903951004, 2.81452384711716862351284369338, 2.99256636265211871951421339073, 3.00828099489773107372633597923, 3.59945666760147367008141919173, 3.65230448601682388537914482538, 3.77630586526435975535751320334, 4.05103705666245124434027104217, 4.66611721763567886754645592883, 4.76365493224118229097579702390, 4.84872088774248627400816145810, 4.85832966913033718092105022409, 5.21559152981152156899512728043, 5.35697581715995615644580918662, 5.55788632991090118603725359419, 5.63357519933775621292865858143, 6.07187244337707152199807554226, 6.25199012258728612563505704159, 6.31759477770257476063171847735, 6.71902792154083534705674743420, 7.08132901188523211428703620602
