L(s) = 1 | + 2-s − 3-s + 4-s + 0.332·5-s − 6-s + 7-s + 8-s + 9-s + 0.332·10-s + 0.0285·11-s − 12-s − 6.71·13-s + 14-s − 0.332·15-s + 16-s + 3.25·17-s + 18-s + 1.72·19-s + 0.332·20-s − 21-s + 0.0285·22-s + 0.0708·23-s − 24-s − 4.88·25-s − 6.71·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.148·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.105·10-s + 0.00862·11-s − 0.288·12-s − 1.86·13-s + 0.267·14-s − 0.0858·15-s + 0.250·16-s + 0.790·17-s + 0.235·18-s + 0.396·19-s + 0.0743·20-s − 0.218·21-s + 0.00609·22-s + 0.0147·23-s − 0.204·24-s − 0.977·25-s − 1.31·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.627567449\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.627567449\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
good | 5 | \( 1 - 0.332T + 5T^{2} \) |
| 11 | \( 1 - 0.0285T + 11T^{2} \) |
| 13 | \( 1 + 6.71T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 - 1.72T + 19T^{2} \) |
| 23 | \( 1 - 0.0708T + 23T^{2} \) |
| 29 | \( 1 - 0.0993T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 - 0.999T + 41T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 + 3.45T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 - 7.83T + 61T^{2} \) |
| 67 | \( 1 + 0.0586T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 4.61T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.56157800627766911398095311449, −7.19779188543359781099066812082, −6.25782466738329109763780900055, −5.66802291807656852866948465854, −4.98563841542614132386885804108, −4.54083210034248681522543182057, −3.61785355369797058473658634635, −2.65801840093709720570395001832, −1.93976781235299917290616413145, −0.73427981102896019959799659275,
0.73427981102896019959799659275, 1.93976781235299917290616413145, 2.65801840093709720570395001832, 3.61785355369797058473658634635, 4.54083210034248681522543182057, 4.98563841542614132386885804108, 5.66802291807656852866948465854, 6.25782466738329109763780900055, 7.19779188543359781099066812082, 7.56157800627766911398095311449