L(s) = 1 | − 2-s + 3-s + 4-s − 1.18·5-s − 6-s + 7-s − 8-s + 9-s + 1.18·10-s − 4.61·11-s + 12-s − 4.97·13-s − 14-s − 1.18·15-s + 16-s + 0.340·17-s − 18-s + 0.525·19-s − 1.18·20-s + 21-s + 4.61·22-s − 5.96·23-s − 24-s − 3.59·25-s + 4.97·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.529·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.374·10-s − 1.39·11-s + 0.288·12-s − 1.38·13-s − 0.267·14-s − 0.305·15-s + 0.250·16-s + 0.0826·17-s − 0.235·18-s + 0.120·19-s − 0.264·20-s + 0.218·21-s + 0.984·22-s − 1.24·23-s − 0.204·24-s − 0.719·25-s + 0.976·26-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
\(L(1)\) |
\(\approx\) |
\(0.9051565225\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9051565225\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.18T + 5T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 0.340T + 17T^{2} \) |
| 19 | \( 1 - 0.525T + 19T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 - 4.88T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 6.02T + 43T^{2} \) |
| 47 | \( 1 - 5.85T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 9.25T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 0.289T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.87243278935945022239103784530, −7.56307564626125270260115121802, −6.72540439581978142019760622906, −5.74964774186675977102066067848, −5.01042464529383573555387809514, −4.26642206562048496320381217671, −3.32607752930182705373692543258, −2.45843062902183940985950974846, −1.96016437581264864654661339066, −0.47921346588279594538764969858,
0.47921346588279594538764969858, 1.96016437581264864654661339066, 2.45843062902183940985950974846, 3.32607752930182705373692543258, 4.26642206562048496320381217671, 5.01042464529383573555387809514, 5.74964774186675977102066067848, 6.72540439581978142019760622906, 7.56307564626125270260115121802, 7.87243278935945022239103784530