| L(s) = 1 | − 10·7-s − 2·13-s − 58·19-s − 22·25-s + 20·31-s − 50·37-s − 28·43-s − 23·49-s + 46·61-s + 38·67-s − 194·73-s − 154·79-s + 20·91-s − 98·97-s + 326·103-s + 4·109-s + 170·121-s + 127-s + 131-s + 580·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.42·7-s − 0.153·13-s − 3.05·19-s − 0.879·25-s + 0.645·31-s − 1.35·37-s − 0.651·43-s − 0.469·49-s + 0.754·61-s + 0.567·67-s − 2.65·73-s − 1.94·79-s + 0.219·91-s − 1.01·97-s + 3.16·103-s + 0.0366·109-s + 1.40·121-s + 0.00787·127-s + 0.00763·131-s + 4.36·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3792762915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3792762915\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 22 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 170 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 70 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 29 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 986 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4346 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3026 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1750 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 19 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 286 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 77 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 334 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10010 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
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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
−11.26112355675301296660854416515, −10.39882917423926778229311593597, −10.38551402212593264633800543424, −9.874278348910024810992459203828, −9.481408124847259929036453052173, −8.752783526818138391610687219170, −8.527364470041752726914234172700, −8.174977818251277283358721579044, −7.28756510786691398101426957723, −6.91532044676116506355363965209, −6.50111486854269244608566552961, −5.98394248642552994955765472133, −5.72029550466723488285410984144, −4.58223512131180436667074250322, −4.47151566531477322745282370969, −3.60908763348325218518119322586, −3.18152494135739185791584059326, −2.34089481652129161661525775468, −1.75856760444049636494739104216, −0.24645094860029743996447305861,
0.24645094860029743996447305861, 1.75856760444049636494739104216, 2.34089481652129161661525775468, 3.18152494135739185791584059326, 3.60908763348325218518119322586, 4.47151566531477322745282370969, 4.58223512131180436667074250322, 5.72029550466723488285410984144, 5.98394248642552994955765472133, 6.50111486854269244608566552961, 6.91532044676116506355363965209, 7.28756510786691398101426957723, 8.174977818251277283358721579044, 8.527364470041752726914234172700, 8.752783526818138391610687219170, 9.481408124847259929036453052173, 9.874278348910024810992459203828, 10.38551402212593264633800543424, 10.39882917423926778229311593597, 11.26112355675301296660854416515
