L(s) = 1 | − 0.0822·2-s − 1.81·3-s − 1.99·4-s + 0.149·6-s − 0.690·7-s + 0.328·8-s + 0.306·9-s − 2.95·11-s + 3.62·12-s + 1.93·13-s + 0.0567·14-s + 3.95·16-s + 2.07·17-s − 0.0251·18-s + 3.74·19-s + 1.25·21-s + 0.243·22-s − 4.34·23-s − 0.597·24-s − 0.159·26-s + 4.89·27-s + 1.37·28-s − 8.10·29-s − 2.80·31-s − 0.982·32-s + 5.37·33-s − 0.170·34-s + ⋯ |
L(s) = 1 | − 0.0581·2-s − 1.04·3-s − 0.996·4-s + 0.0610·6-s − 0.261·7-s + 0.116·8-s + 0.102·9-s − 0.891·11-s + 1.04·12-s + 0.536·13-s + 0.0151·14-s + 0.989·16-s + 0.503·17-s − 0.00593·18-s + 0.859·19-s + 0.273·21-s + 0.0518·22-s − 0.906·23-s − 0.121·24-s − 0.0312·26-s + 0.942·27-s + 0.260·28-s − 1.50·29-s − 0.503·31-s − 0.173·32-s + 0.936·33-s − 0.0292·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4361449315\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4361449315\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 0.0822T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 + 0.690T + 7T^{2} \) |
| 11 | \( 1 + 2.95T + 11T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 - 9.72T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 - 0.0484T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 - 0.964T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 - 5.25T + 97T^{2} \) |
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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
−7.994285912829420404434200353690, −7.53376795598248593943780450105, −6.42666691474509262546480282074, −5.74021054611252460355309859449, −5.35269905130590820873511060968, −4.63365770378520523492760128568, −3.72232607359106924863736632685, −2.97970321794572061979353450171, −1.53050716002621438157860898574, −0.38475321813359077540418061694,
0.38475321813359077540418061694, 1.53050716002621438157860898574, 2.97970321794572061979353450171, 3.72232607359106924863736632685, 4.63365770378520523492760128568, 5.35269905130590820873511060968, 5.74021054611252460355309859449, 6.42666691474509262546480282074, 7.53376795598248593943780450105, 7.994285912829420404434200353690