| L(s) = 1 | + 5·2-s + 7·4-s + 15·5-s + 15·7-s − 15·8-s + 75·10-s − 17·11-s + 75·14-s − 65·16-s − 70·17-s − 141·19-s + 105·20-s − 85·22-s − 145·23-s + 25·25-s + 105·28-s − 34·29-s − 140·31-s − 95·32-s − 350·34-s + 225·35-s − 190·37-s − 705·38-s − 225·40-s − 538·41-s + 455·43-s − 119·44-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 7/8·4-s + 1.34·5-s + 0.809·7-s − 0.662·8-s + 2.37·10-s − 0.465·11-s + 1.43·14-s − 1.01·16-s − 0.998·17-s − 1.70·19-s + 1.17·20-s − 0.823·22-s − 1.31·23-s + 1/5·25-s + 0.708·28-s − 0.217·29-s − 0.811·31-s − 0.524·32-s − 1.76·34-s + 1.08·35-s − 0.844·37-s − 3.00·38-s − 0.889·40-s − 2.04·41-s + 1.61·43-s − 0.407·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 5 T + 9 p T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 p T + 8 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 15 T + 738 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 17 T + 1778 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 70 T + 9963 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 141 T + 978 p T^{2} + 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 145 T + 24962 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 34 T + 33767 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 140 T + 21982 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 190 T + 102103 T^{2} + 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 538 T + 208503 T^{2} + 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 455 T + 170782 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 60 T + 125246 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 545 T + 256304 T^{2} + 545 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 809 T + 561522 T^{2} - 809 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 502 T + 346963 T^{2} - 502 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 475 T + 622738 T^{2} + 475 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 127 T + 672998 T^{2} + 127 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 585 T + 832884 T^{2} - 585 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 240 T + 993678 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 260 T + 1117974 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 921 T + 1555592 T^{2} + 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 415 T + 933568 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.799722706409645660572593637243, −8.452744697228628296318353397802, −8.219434246748475562559429762658, −7.73512026242645426103144069223, −6.87476277329832332185646800392, −6.81244566606679427048644429924, −6.14569818452916036340595705413, −5.93536270318192295273082171192, −5.38702614432198955271145704166, −5.26506729296925147544531792352, −4.58137958742649290985586532051, −4.51205546209074883089296415426, −3.73917704178898812906298892589, −3.69199647175063466005914692404, −2.73837992008216085097419852136, −2.28497017755463231702982176320, −1.85421647489021626762030370602, −1.49815554504421215419380241640, 0, 0,
1.49815554504421215419380241640, 1.85421647489021626762030370602, 2.28497017755463231702982176320, 2.73837992008216085097419852136, 3.69199647175063466005914692404, 3.73917704178898812906298892589, 4.51205546209074883089296415426, 4.58137958742649290985586532051, 5.26506729296925147544531792352, 5.38702614432198955271145704166, 5.93536270318192295273082171192, 6.14569818452916036340595705413, 6.81244566606679427048644429924, 6.87476277329832332185646800392, 7.73512026242645426103144069223, 8.219434246748475562559429762658, 8.452744697228628296318353397802, 8.799722706409645660572593637243
