| L(s) = 1 | − 2-s + 2·4-s − 2·5-s + 4·7-s − 5·8-s + 2·10-s − 4·11-s − 13-s − 4·14-s + 5·16-s + 4·17-s − 4·20-s + 4·22-s + 5·25-s + 26-s + 8·28-s + 10·29-s − 4·31-s − 10·32-s − 4·34-s − 8·35-s − 4·37-s + 10·40-s − 6·41-s + 12·43-s − 8·44-s + 7·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 0.894·5-s + 1.51·7-s − 1.76·8-s + 0.632·10-s − 1.20·11-s − 0.277·13-s − 1.06·14-s + 5/4·16-s + 0.970·17-s − 0.894·20-s + 0.852·22-s + 25-s + 0.196·26-s + 1.51·28-s + 1.85·29-s − 0.718·31-s − 1.76·32-s − 0.685·34-s − 1.35·35-s − 0.657·37-s + 1.58·40-s − 0.937·41-s + 1.82·43-s − 1.20·44-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214956093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214956093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.25697650412863615088700128720, −9.799815301807426672559061395746, −9.096259295277297379348565894476, −8.663604449681589915151562912100, −8.599251330176785936745951053385, −7.961657803175276911099579320558, −7.63760159610305741771359912362, −7.52120848754557787243523229287, −6.81373758955691357679580602282, −6.55300868852639826595281104303, −5.70556292082506757176595982950, −5.50318358385417967926218792156, −4.97656881435697260778614980774, −4.52850970604890807957077009337, −3.80735421179063566397293340183, −3.21106786701521275019249777931, −2.69980677707273842439706379571, −2.26798651304069144346399973742, −1.40526707573869599306845730655, −0.57784854588684380191294648504,
0.57784854588684380191294648504, 1.40526707573869599306845730655, 2.26798651304069144346399973742, 2.69980677707273842439706379571, 3.21106786701521275019249777931, 3.80735421179063566397293340183, 4.52850970604890807957077009337, 4.97656881435697260778614980774, 5.50318358385417967926218792156, 5.70556292082506757176595982950, 6.55300868852639826595281104303, 6.81373758955691357679580602282, 7.52120848754557787243523229287, 7.63760159610305741771359912362, 7.961657803175276911099579320558, 8.599251330176785936745951053385, 8.663604449681589915151562912100, 9.096259295277297379348565894476, 9.799815301807426672559061395746, 10.25697650412863615088700128720
