L(s) = 1 | + 1.97·2-s − 0.955·3-s + 1.90·4-s + 3.57·5-s − 1.88·6-s + 2.56·7-s − 0.191·8-s − 2.08·9-s + 7.06·10-s − 2.18·11-s − 1.81·12-s + 1.23·13-s + 5.07·14-s − 3.41·15-s − 4.18·16-s + 2.68·17-s − 4.12·18-s + 8.30·19-s + 6.80·20-s − 2.45·21-s − 4.32·22-s + 1.65·23-s + 0.183·24-s + 7.79·25-s + 2.43·26-s + 4.86·27-s + 4.88·28-s + ⋯ |
L(s) = 1 | + 1.39·2-s − 0.551·3-s + 0.951·4-s + 1.59·5-s − 0.770·6-s + 0.970·7-s − 0.0677·8-s − 0.695·9-s + 2.23·10-s − 0.659·11-s − 0.525·12-s + 0.341·13-s + 1.35·14-s − 0.882·15-s − 1.04·16-s + 0.650·17-s − 0.971·18-s + 1.90·19-s + 1.52·20-s − 0.535·21-s − 0.921·22-s + 0.344·23-s + 0.0373·24-s + 1.55·25-s + 0.477·26-s + 0.935·27-s + 0.923·28-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
\(L(1)\) |
\(\approx\) |
\(4.185694960\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.185694960\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 + 0.955T + 3T^{2} \) |
| 5 | \( 1 - 3.57T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 - 8.30T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.61T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 - 3.99T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 2.17T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 + 5.37T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 - 7.02T + 83T^{2} \) |
| 89 | \( 1 + 9.42T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−9.353644461450891209461783638433, −8.478031081361705536252259657019, −7.41343125094018982170407548548, −6.35789852077102338557358745448, −5.66715085457959336377021274546, −5.30290068547691390319174592682, −4.75627429187205668327134640796, −3.27696391298969689795978066729, −2.54805523161992435098731333632, −1.31598956612876023637526308771,
1.31598956612876023637526308771, 2.54805523161992435098731333632, 3.27696391298969689795978066729, 4.75627429187205668327134640796, 5.30290068547691390319174592682, 5.66715085457959336377021274546, 6.35789852077102338557358745448, 7.41343125094018982170407548548, 8.478031081361705536252259657019, 9.353644461450891209461783638433