L(s) = 1 | + 2-s − 0.771·3-s + 4-s − 0.437·5-s − 0.771·6-s − 4.61·7-s + 8-s − 2.40·9-s − 0.437·10-s − 1.80·11-s − 0.771·12-s + 6.12·13-s − 4.61·14-s + 0.336·15-s + 16-s − 2.20·17-s − 2.40·18-s + 1.36·19-s − 0.437·20-s + 3.55·21-s − 1.80·22-s − 0.264·23-s − 0.771·24-s − 4.80·25-s + 6.12·26-s + 4.16·27-s − 4.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.445·3-s + 0.5·4-s − 0.195·5-s − 0.314·6-s − 1.74·7-s + 0.353·8-s − 0.801·9-s − 0.138·10-s − 0.543·11-s − 0.222·12-s + 1.69·13-s − 1.23·14-s + 0.0870·15-s + 0.250·16-s − 0.534·17-s − 0.566·18-s + 0.312·19-s − 0.0977·20-s + 0.775·21-s − 0.384·22-s − 0.0552·23-s − 0.157·24-s − 0.961·25-s + 1.20·26-s + 0.802·27-s − 0.871·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.575553102\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.575553102\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + 0.771T + 3T^{2} \) |
| 5 | \( 1 + 0.437T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 + 2.20T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 0.264T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 - 6.80T + 31T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 6.57T + 47T^{2} \) |
| 53 | \( 1 - 8.81T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.60T + 61T^{2} \) |
| 67 | \( 1 - 2.02T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.17T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 8.24T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 4.11T + 97T^{2} \) |
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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
−8.755845246222323701934421960807, −8.097113451002843524744468211709, −6.95604846642844338892792011566, −6.21093182574989105892561052022, −6.01579724187601449994137720121, −5.01220268393543167492852173559, −3.88492172926377264681767876217, −3.28841405870470454045920994450, −2.47048760289464139110962068932, −0.69149607506668768568094277197,
0.69149607506668768568094277197, 2.47048760289464139110962068932, 3.28841405870470454045920994450, 3.88492172926377264681767876217, 5.01220268393543167492852173559, 6.01579724187601449994137720121, 6.21093182574989105892561052022, 6.95604846642844338892792011566, 8.097113451002843524744468211709, 8.755845246222323701934421960807