| L(s) = 1 | − 4·3-s + 4·5-s + 8·9-s − 8·13-s − 16·15-s − 12·17-s + 16·23-s + 2·25-s − 12·27-s − 24·29-s − 8·31-s − 4·37-s + 32·39-s − 8·43-s + 32·45-s + 48·51-s + 24·53-s − 16·59-s − 16·61-s − 32·65-s + 16·67-s − 64·69-s + 8·73-s − 8·75-s + 23·81-s − 16·83-s − 48·85-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 8/3·9-s − 2.21·13-s − 4.13·15-s − 2.91·17-s + 3.33·23-s + 2/5·25-s − 2.30·27-s − 4.45·29-s − 1.43·31-s − 0.657·37-s + 5.12·39-s − 1.21·43-s + 4.77·45-s + 6.72·51-s + 3.29·53-s − 2.08·59-s − 2.04·61-s − 3.96·65-s + 1.95·67-s − 7.70·69-s + 0.936·73-s − 0.923·75-s + 23/9·81-s − 1.75·83-s − 5.20·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4893981414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4893981414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1726 T^{4} + 372 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 414 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 816 T^{3} + 4418 T^{4} - 816 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 140 T^{3} + 2446 T^{4} + 140 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 392 T^{3} - 3662 T^{4} - 392 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 4382 T^{4} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2904 T^{3} + 24658 T^{4} - 2904 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 7922 T^{4} - 1008 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 152 T^{2} + 12658 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 616 T^{3} + 11842 T^{4} - 616 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 22534 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 234 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 3192 T^{3} + 34082 T^{4} - 3192 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.17675599506848478855438277261, −7.05629250780253093806865800704, −6.97564466319982997348679362061, −6.38664762896261384807462538152, −6.30833422099189095251498427031, −6.17733375753313275128495849565, −6.11280040044392796771447259654, −5.42465348026794968706521212787, −5.35859324092417607819055618152, −5.33413044486938646293777392573, −5.25383249944260467993968126252, −4.83439160406069197406487690984, −4.74532196998149119705703727424, −4.43927757150735723485622527471, −3.91565553381707999709323880296, −3.80192416190551237329927135778, −3.51205383964035603136949826439, −3.00172951442939260051574880925, −2.48423203603451937044275241281, −2.44063465019774536116298168154, −1.85860141477612220667565952223, −1.78310694600286166972684332118, −1.71662192667073706455546001342, −0.59272456669172662976795701830, −0.30729335538140395282626972200,
0.30729335538140395282626972200, 0.59272456669172662976795701830, 1.71662192667073706455546001342, 1.78310694600286166972684332118, 1.85860141477612220667565952223, 2.44063465019774536116298168154, 2.48423203603451937044275241281, 3.00172951442939260051574880925, 3.51205383964035603136949826439, 3.80192416190551237329927135778, 3.91565553381707999709323880296, 4.43927757150735723485622527471, 4.74532196998149119705703727424, 4.83439160406069197406487690984, 5.25383249944260467993968126252, 5.33413044486938646293777392573, 5.35859324092417607819055618152, 5.42465348026794968706521212787, 6.11280040044392796771447259654, 6.17733375753313275128495849565, 6.30833422099189095251498427031, 6.38664762896261384807462538152, 6.97564466319982997348679362061, 7.05629250780253093806865800704, 7.17675599506848478855438277261
