L(s) = 1 | − 2.60·2-s + 2.97·3-s + 4.77·4-s + 0.851·5-s − 7.75·6-s + 3.04·7-s − 7.23·8-s + 5.87·9-s − 2.21·10-s − 11-s + 14.2·12-s + 1.25·13-s − 7.92·14-s + 2.53·15-s + 9.28·16-s − 7.73·17-s − 15.2·18-s + 6.82·19-s + 4.06·20-s + 9.06·21-s + 2.60·22-s + 8.40·23-s − 21.5·24-s − 4.27·25-s − 3.27·26-s + 8.56·27-s + 14.5·28-s + ⋯ |
L(s) = 1 | − 1.84·2-s + 1.71·3-s + 2.38·4-s + 0.380·5-s − 3.16·6-s + 1.15·7-s − 2.55·8-s + 1.95·9-s − 0.701·10-s − 0.301·11-s + 4.11·12-s + 0.348·13-s − 2.11·14-s + 0.654·15-s + 2.32·16-s − 1.87·17-s − 3.60·18-s + 1.56·19-s + 0.909·20-s + 1.97·21-s + 0.555·22-s + 1.75·23-s − 4.40·24-s − 0.855·25-s − 0.641·26-s + 1.64·27-s + 2.74·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.434699284\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.434699284\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 - 0.851T + 5T^{2} \) |
| 7 | \( 1 - 3.04T + 7T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 8.40T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 + 9.82T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 4.72T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 + 7.00T + 71T^{2} \) |
| 73 | \( 1 - 1.27T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 3.62T + 89T^{2} \) |
| 97 | \( 1 + 0.113T + 97T^{2} \) |
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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
−10.11247617120883274978548505092, −9.368983541366502148078643964522, −8.682121105489843642237222553735, −8.317892571957211476011345106899, −7.39520946576578936140901741198, −6.80705890993076178194043429334, −5.01629584045294775542968488282, −3.31834160293547262233947083900, −2.22326558543756759133381733443, −1.48630621870212293968634266202,
1.48630621870212293968634266202, 2.22326558543756759133381733443, 3.31834160293547262233947083900, 5.01629584045294775542968488282, 6.80705890993076178194043429334, 7.39520946576578936140901741198, 8.317892571957211476011345106899, 8.682121105489843642237222553735, 9.368983541366502148078643964522, 10.11247617120883274978548505092