| L(s) = 1 | + 12·7-s + 4·13-s − 76·19-s − 10·25-s − 36·31-s − 160·37-s + 44·43-s − 16·49-s + 76·61-s − 216·67-s − 92·73-s + 300·79-s + 48·91-s + 584·97-s + 208·103-s + 284·109-s + 216·121-s + 127-s + 131-s − 912·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 12/7·7-s + 4/13·13-s − 4·19-s − 2/5·25-s − 1.16·31-s − 4.32·37-s + 1.02·43-s − 0.326·49-s + 1.24·61-s − 3.22·67-s − 1.26·73-s + 3.79·79-s + 0.527·91-s + 6.02·97-s + 2.01·103-s + 2.60·109-s + 1.78·121-s + 0.00787·127-s + 0.00763·131-s − 6.85·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.007247385089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007247385089\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 - 6 T + 62 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 36446 T^{4} - 216 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 294 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 978 T^{2} + 403283 T^{4} - 978 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 p T + 903 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1306 T^{2} + 824091 T^{4} - 1306 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 216 T^{2} - 1234 p^{2} T^{4} - 216 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 18 T + 383 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 80 T + 4158 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 1224 T^{2} + 5946686 T^{4} + 1224 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 22 T + 174 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1788 T^{2} + 8611718 T^{4} - 1788 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 5034 T^{2} + 16914251 T^{4} - 5034 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3384 T^{2} + 26924606 T^{4} + 3384 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 19 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_{4}$ | \( ( 1 + 108 T + 9014 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 7896 T^{2} + 66189566 T^{4} - 7896 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 46 T + 5742 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 150 T + 17387 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 21634 T^{2} + 210991011 T^{4} - 21634 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23656 T^{2} + 264671646 T^{4} - 23656 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 292 T + 39954 T^{2} - 292 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.11161845356927546343366016662, −6.10205251095580528797401230094, −6.04007744796639239339010210227, −5.72431975811020245682961968459, −5.30141958087082124374897291873, −5.05033298304177720716392426793, −5.01818426196790858259961512434, −4.65556504414657407545224111405, −4.62982141710955323171616422141, −4.49313140391637631280704410714, −4.21407023121344698103892943896, −3.74631550138165721785105844891, −3.63280684320194410834996059613, −3.53349827680724688223028379100, −3.28645981873016827030365947993, −3.00186938967846543844652063319, −2.45336577399255553116952160960, −2.06377918535528321703832124314, −2.02547604838880961695187364923, −1.97375037766550108375594525580, −1.73397969978561061429000774370, −1.49039575996082607417844190646, −0.72420722544954815431662929306, −0.67051896181243279976574366335, −0.01067148360101303833197594064,
0.01067148360101303833197594064, 0.67051896181243279976574366335, 0.72420722544954815431662929306, 1.49039575996082607417844190646, 1.73397969978561061429000774370, 1.97375037766550108375594525580, 2.02547604838880961695187364923, 2.06377918535528321703832124314, 2.45336577399255553116952160960, 3.00186938967846543844652063319, 3.28645981873016827030365947993, 3.53349827680724688223028379100, 3.63280684320194410834996059613, 3.74631550138165721785105844891, 4.21407023121344698103892943896, 4.49313140391637631280704410714, 4.62982141710955323171616422141, 4.65556504414657407545224111405, 5.01818426196790858259961512434, 5.05033298304177720716392426793, 5.30141958087082124374897291873, 5.72431975811020245682961968459, 6.04007744796639239339010210227, 6.10205251095580528797401230094, 6.11161845356927546343366016662
