L(s) = 1 | − 2.17·2-s − 0.327·3-s + 2.72·4-s + 0.0263·5-s + 0.712·6-s + 3.12·7-s − 1.57·8-s − 2.89·9-s − 0.0572·10-s − 3.73·11-s − 0.893·12-s − 4.89·13-s − 6.80·14-s − 0.00862·15-s − 2.02·16-s + 5.91·17-s + 6.28·18-s − 3.96·19-s + 0.0717·20-s − 1.02·21-s + 8.12·22-s + 0.343·23-s + 0.517·24-s − 4.99·25-s + 10.6·26-s + 1.93·27-s + 8.53·28-s + ⋯ |
L(s) = 1 | − 1.53·2-s − 0.189·3-s + 1.36·4-s + 0.0117·5-s + 0.290·6-s + 1.18·7-s − 0.558·8-s − 0.964·9-s − 0.0180·10-s − 1.12·11-s − 0.257·12-s − 1.35·13-s − 1.81·14-s − 0.00222·15-s − 0.505·16-s + 1.43·17-s + 1.48·18-s − 0.910·19-s + 0.0160·20-s − 0.223·21-s + 1.73·22-s + 0.0717·23-s + 0.105·24-s − 0.999·25-s + 2.08·26-s + 0.371·27-s + 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5625038003\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5625038003\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 3 | \( 1 + 0.327T + 3T^{2} \) |
| 5 | \( 1 - 0.0263T + 5T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 - 0.343T + 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 5.81T + 53T^{2} \) |
| 59 | \( 1 - 0.172T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 79 | \( 1 + 5.41T + 79T^{2} \) |
| 83 | \( 1 - 2.03T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−9.243881574802470139176279741171, −8.293514330700771376181563525955, −7.84538150760321247441883124449, −7.46602531297803226209200196130, −6.14210185299696874220926711727, −5.28929380466886669998599791385, −4.50231275131341818074642858377, −2.76764195207913969282257331007, −2.04013240937753539197734578312, −0.62775645065121058602318580736,
0.62775645065121058602318580736, 2.04013240937753539197734578312, 2.76764195207913969282257331007, 4.50231275131341818074642858377, 5.28929380466886669998599791385, 6.14210185299696874220926711727, 7.46602531297803226209200196130, 7.84538150760321247441883124449, 8.293514330700771376181563525955, 9.243881574802470139176279741171