L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 1.41·5-s − 6·6-s − 4.62·7-s − 8·8-s + 9·9-s + 2.83·10-s + 10.6·11-s + 12·12-s − 55.5·13-s + 9.25·14-s − 4.25·15-s + 16·16-s − 53.7·17-s − 18·18-s + 32.4·19-s − 5.66·20-s − 13.8·21-s − 21.3·22-s + 39.6·23-s − 24·24-s − 122.·25-s + 111.·26-s + 27·27-s − 18.5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.126·5-s − 0.408·6-s − 0.249·7-s − 0.353·8-s + 0.333·9-s + 0.0896·10-s + 0.292·11-s + 0.288·12-s − 1.18·13-s + 0.176·14-s − 0.0731·15-s + 0.250·16-s − 0.767·17-s − 0.235·18-s + 0.391·19-s − 0.0633·20-s − 0.144·21-s − 0.206·22-s + 0.359·23-s − 0.204·24-s − 0.983·25-s + 0.838·26-s + 0.192·27-s − 0.124·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
good | 5 | \( 1 + 1.41T + 125T^{2} \) |
| 7 | \( 1 + 4.62T + 343T^{2} \) |
| 11 | \( 1 - 10.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 84.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 84.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 422.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 189.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 670.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 95.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 531.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 472.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 586.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 783.T + 9.12e5T^{2} \) |
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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
−10.39842312235092166747888086393, −9.434524225157456034571411396871, −8.950346972683913039652539673201, −7.66877511565074294193236215581, −7.15350353844025734933875426588, −5.84471775636314025039184516386, −4.38096612217478391948785343793, −3.03724418876622276416973931102, −1.82705074670723802525610652131, 0,
1.82705074670723802525610652131, 3.03724418876622276416973931102, 4.38096612217478391948785343793, 5.84471775636314025039184516386, 7.15350353844025734933875426588, 7.66877511565074294193236215581, 8.950346972683913039652539673201, 9.434524225157456034571411396871, 10.39842312235092166747888086393