| L(s) = 1 | + 4·4-s − 12·5-s + 12·16-s + 12·17-s − 12·19-s − 48·20-s − 60·23-s + 66·25-s − 68·31-s + 240·47-s + 8·49-s − 204·53-s − 196·61-s + 32·64-s + 48·68-s − 48·76-s + 180·79-s − 144·80-s − 108·83-s − 144·85-s − 240·92-s + 144·95-s + 264·100-s + 144·107-s − 76·109-s − 48·113-s + 720·115-s + ⋯ |
| L(s) = 1 | + 4-s − 2.39·5-s + 3/4·16-s + 0.705·17-s − 0.631·19-s − 2.39·20-s − 2.60·23-s + 2.63·25-s − 2.19·31-s + 5.10·47-s + 8/49·49-s − 3.84·53-s − 3.21·61-s + 1/2·64-s + 0.705·68-s − 0.631·76-s + 2.27·79-s − 9/5·80-s − 1.30·83-s − 1.69·85-s − 2.60·92-s + 1.51·95-s + 2.63·100-s + 1.34·107-s − 0.697·109-s − 0.424·113-s + 6.26·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.003093119092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003093119092\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 7 | $C_2^2 \wr C_2$ | \( 1 - 8 T^{2} - 3894 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 96 T^{2} + 29786 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 352 T^{2} + 71898 T^{4} - 352 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 489 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 713 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 30 T + 891 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1096 T^{2} + 1242474 T^{4} - 1096 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 34 T + 1761 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 5352 T^{2} + 12407498 T^{4} - 5352 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 5672 T^{2} + 14203050 T^{4} - 5672 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 120 T + 7626 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 102 T + 8057 T^{2} + 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 10848 T^{2} + 53616410 T^{4} - 10848 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 98 T + 8691 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 6508 T^{2} + 32310150 T^{4} + 6508 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 90 T + 6569 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 54 T + 7779 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 19172 T^{2} + 267913158 T^{4} - 19172 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−8.417304005703146287284569980597, −8.062347681090796004110683855964, −7.72587265447779194131888066790, −7.70160775134969670698516823934, −7.51181083065864069572447934547, −7.35781161800555583840081507828, −7.05306337822045578066539765849, −6.61995464469389639668984929066, −6.23497802634086679275257256508, −6.18683368950351144786000148334, −5.81820864562338701314150277122, −5.47586346339973710203501027583, −5.35918875587960436062511752831, −4.55627242451186463850779747897, −4.49266772781826075664787255832, −4.19778619062035900273620195074, −3.94615722225874517332502935570, −3.46828571321495591781688823344, −3.42304134135874615096236568517, −3.03322616900163626894039551733, −2.39408445534723344321305199832, −2.09405665828087729871698837057, −1.65220395114889609306075717394, −0.963257331480092248079427387955, −0.01546708143175349743667653317,
0.01546708143175349743667653317, 0.963257331480092248079427387955, 1.65220395114889609306075717394, 2.09405665828087729871698837057, 2.39408445534723344321305199832, 3.03322616900163626894039551733, 3.42304134135874615096236568517, 3.46828571321495591781688823344, 3.94615722225874517332502935570, 4.19778619062035900273620195074, 4.49266772781826075664787255832, 4.55627242451186463850779747897, 5.35918875587960436062511752831, 5.47586346339973710203501027583, 5.81820864562338701314150277122, 6.18683368950351144786000148334, 6.23497802634086679275257256508, 6.61995464469389639668984929066, 7.05306337822045578066539765849, 7.35781161800555583840081507828, 7.51181083065864069572447934547, 7.70160775134969670698516823934, 7.72587265447779194131888066790, 8.062347681090796004110683855964, 8.417304005703146287284569980597
