| L(s) = 1 | + 2-s − 3-s + 4-s − 0.831·5-s − 6-s + 8-s + 9-s − 0.831·10-s − 2.45·11-s − 12-s − 5.30·13-s + 0.831·15-s + 16-s − 3.28·17-s + 18-s − 2·19-s − 0.831·20-s − 2.45·22-s − 23-s − 24-s − 4.30·25-s − 5.30·26-s − 27-s + 10.3·29-s + 0.831·30-s − 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.371·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.263·10-s − 0.740·11-s − 0.288·12-s − 1.47·13-s + 0.214·15-s + 0.250·16-s − 0.797·17-s + 0.235·18-s − 0.458·19-s − 0.185·20-s − 0.523·22-s − 0.208·23-s − 0.204·24-s − 0.861·25-s − 1.04·26-s − 0.192·27-s + 1.91·29-s + 0.151·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564691558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564691558\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 0.831T + 5T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2.33T + 37T^{2} \) |
| 41 | \( 1 - 3.66T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3.91T + 47T^{2} \) |
| 53 | \( 1 + 7.78T + 53T^{2} \) |
| 59 | \( 1 + 3.32T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 6.57T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.84815797537808526382290971653, −7.11924837957014566217018083398, −6.54684629396375855424926976296, −5.78073344562488058485746025209, −4.98423284336784908639812922676, −4.56770176504439449161409550323, −3.78868931317032603761828967680, −2.65962650529489541107923388479, −2.12514164225991794548145722955, −0.56131630186344711461171840553,
0.56131630186344711461171840553, 2.12514164225991794548145722955, 2.65962650529489541107923388479, 3.78868931317032603761828967680, 4.56770176504439449161409550323, 4.98423284336784908639812922676, 5.78073344562488058485746025209, 6.54684629396375855424926976296, 7.11924837957014566217018083398, 7.84815797537808526382290971653
