L(s) = 1 | + 2-s − 3-s + 4-s + 1.04·5-s − 6-s − 0.892·7-s + 8-s + 9-s + 1.04·10-s − 2.88·11-s − 12-s + 13-s − 0.892·14-s − 1.04·15-s + 16-s + 5.24·17-s + 18-s − 3.82·19-s + 1.04·20-s + 0.892·21-s − 2.88·22-s + 3.40·23-s − 24-s − 3.89·25-s + 26-s − 27-s − 0.892·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.469·5-s − 0.408·6-s − 0.337·7-s + 0.353·8-s + 0.333·9-s + 0.331·10-s − 0.870·11-s − 0.288·12-s + 0.277·13-s − 0.238·14-s − 0.270·15-s + 0.250·16-s + 1.27·17-s + 0.235·18-s − 0.878·19-s + 0.234·20-s + 0.194·21-s − 0.615·22-s + 0.710·23-s − 0.204·24-s − 0.779·25-s + 0.196·26-s − 0.192·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.728682033\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.728682033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 + 0.892T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 3.40T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 3.82T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 2.99T + 41T^{2} \) |
| 43 | \( 1 - 0.469T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 5.26T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 0.663T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
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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
−7.63526403605268144801764861707, −7.00983394309292857040026336076, −6.16682719510683557479350095263, −5.80912371469425950642039301959, −5.09655456501638669655247946747, −4.44857714932745972369449420912, −3.51377907369539900706664319052, −2.78483976598069477836596366049, −1.88083419318736556095840417263, −0.75480199887115747638491945811,
0.75480199887115747638491945811, 1.88083419318736556095840417263, 2.78483976598069477836596366049, 3.51377907369539900706664319052, 4.44857714932745972369449420912, 5.09655456501638669655247946747, 5.80912371469425950642039301959, 6.16682719510683557479350095263, 7.00983394309292857040026336076, 7.63526403605268144801764861707