L(s) = 1 | − 2-s + 0.772·3-s + 4-s + 1.22·5-s − 0.772·6-s − 3.40·7-s − 8-s − 2.40·9-s − 1.22·10-s + 0.772·12-s + 1.40·13-s + 3.40·14-s + 0.948·15-s + 16-s + 4.80·17-s + 2.40·18-s + 19-s + 1.22·20-s − 2.62·21-s + 6.80·23-s − 0.772·24-s − 3.49·25-s − 1.40·26-s − 4.17·27-s − 3.40·28-s − 8.03·29-s − 0.948·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.446·3-s + 0.5·4-s + 0.548·5-s − 0.315·6-s − 1.28·7-s − 0.353·8-s − 0.800·9-s − 0.388·10-s + 0.223·12-s + 0.389·13-s + 0.909·14-s + 0.244·15-s + 0.250·16-s + 1.16·17-s + 0.566·18-s + 0.229·19-s + 0.274·20-s − 0.573·21-s + 1.41·23-s − 0.157·24-s − 0.698·25-s − 0.275·26-s − 0.803·27-s − 0.643·28-s − 1.49·29-s − 0.173·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.772T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 + 8.03T + 29T^{2} \) |
| 31 | \( 1 + 4.94T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 - 5.25T + 47T^{2} \) |
| 53 | \( 1 + 4.45T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 7.57T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 2.35T + 97T^{2} \) |
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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
−7.903108339144659327807809891536, −7.44487620750161314651211900116, −6.52229166538242133939419240547, −5.83979321928814990869847321724, −5.37299032170386843663735045648, −3.78357062610461840899419784648, −3.19047303599234683116520019577, −2.48093162165497453002001686116, −1.33231216921135747237676385657, 0,
1.33231216921135747237676385657, 2.48093162165497453002001686116, 3.19047303599234683116520019577, 3.78357062610461840899419784648, 5.37299032170386843663735045648, 5.83979321928814990869847321724, 6.52229166538242133939419240547, 7.44487620750161314651211900116, 7.903108339144659327807809891536