L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−1.23 + 1.04i)5-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−1.52 + 0.553i)10-s + (−0.107 + 0.608i)13-s + (0.173 + 0.984i)16-s + (−0.580 − 0.211i)17-s − 18-s − 1.61·20-s + (0.280 − 1.59i)25-s + (−0.309 + 0.535i)26-s + (0.580 − 0.211i)29-s + (−0.173 + 0.984i)32-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−1.23 + 1.04i)5-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−1.52 + 0.553i)10-s + (−0.107 + 0.608i)13-s + (0.173 + 0.984i)16-s + (−0.580 − 0.211i)17-s − 18-s − 1.61·20-s + (0.280 − 1.59i)25-s + (−0.309 + 0.535i)26-s + (0.580 − 0.211i)29-s + (−0.173 + 0.984i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.341981942\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341981942\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.107 - 0.608i)T + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.580 + 0.211i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.61T + T^{2} \) |
| 41 | \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 1.04i)T + (0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.280 + 1.59i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.580 - 0.211i)T + (0.766 + 0.642i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29452315147806604102814674665, −8.930283851372758145363040131434, −8.041334260872681272509252314773, −7.51911956886964515148378333656, −6.65564419732358591193307074789, −6.04390269995690363425303168658, −4.80626947083717748812947982612, −4.10265195795682511731186264309, −3.11369423793324844311584924725, −2.43922853860226976871691112071,
0.801908867278651279463621912622, 2.51360768531416406287295346914, 3.57423955442126152535036262086, 4.28940816588751869370630059941, 5.13946081901521000304474046282, 5.88855857315006309167638803147, 6.93501522156106692906709138561, 7.897846845132654088037882208748, 8.559148092534384642704337328633, 9.409533761398737793214908944936