| L(s) = 1 | − 2·2-s + 3·4-s + 4·5-s − 4·7-s − 4·8-s + 2·9-s − 8·10-s + 8·11-s + 8·14-s + 5·16-s − 4·18-s − 12·19-s + 12·20-s − 16·22-s + 4·25-s − 12·28-s + 8·29-s − 2·31-s − 6·32-s − 16·35-s + 6·36-s − 8·37-s + 24·38-s − 16·40-s + 12·41-s + 24·44-s + 8·45-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.51·7-s − 1.41·8-s + 2/3·9-s − 2.52·10-s + 2.41·11-s + 2.13·14-s + 5/4·16-s − 0.942·18-s − 2.75·19-s + 2.68·20-s − 3.41·22-s + 4/5·25-s − 2.26·28-s + 1.48·29-s − 0.359·31-s − 1.06·32-s − 2.70·35-s + 36-s − 1.31·37-s + 3.89·38-s − 2.52·40-s + 1.87·41-s + 3.61·44-s + 1.19·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36168196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36168196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.751171805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751171805\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| 97 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 128 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 20 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.443189560511459747444609163592, −8.111266969007137768920229800998, −7.41994009968835948346450794809, −6.97455913411906210154583447851, −6.76672936002661745387983850572, −6.62362291719492122175110511244, −6.28310156813118760588953968086, −6.05844581626898164268734798554, −5.66357593552794528464886274751, −5.30831035900583065410417498733, −4.39761959931704293522981786357, −4.16141597684171874711300127116, −3.84614812268352908064876691143, −3.41817717998155416771170855243, −2.58112461273422781519765151843, −2.41396682326167070570363849140, −2.04854126208703496798578991928, −1.44828299760589920255174745968, −1.15560462895476239860559086851, −0.44680072863095070128974502024,
0.44680072863095070128974502024, 1.15560462895476239860559086851, 1.44828299760589920255174745968, 2.04854126208703496798578991928, 2.41396682326167070570363849140, 2.58112461273422781519765151843, 3.41817717998155416771170855243, 3.84614812268352908064876691143, 4.16141597684171874711300127116, 4.39761959931704293522981786357, 5.30831035900583065410417498733, 5.66357593552794528464886274751, 6.05844581626898164268734798554, 6.28310156813118760588953968086, 6.62362291719492122175110511244, 6.76672936002661745387983850572, 6.97455913411906210154583447851, 7.41994009968835948346450794809, 8.111266969007137768920229800998, 8.443189560511459747444609163592
