L(s) = 1 | − 2-s − 2.39·3-s + 4-s − 1.11·5-s + 2.39·6-s + 2.17·7-s − 8-s + 2.75·9-s + 1.11·10-s − 3.08·11-s − 2.39·12-s + 2.44·13-s − 2.17·14-s + 2.67·15-s + 16-s + 4.89·17-s − 2.75·18-s − 4.74·19-s − 1.11·20-s − 5.22·21-s + 3.08·22-s − 3.76·23-s + 2.39·24-s − 3.75·25-s − 2.44·26-s + 0.577·27-s + 2.17·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.38·3-s + 0.5·4-s − 0.498·5-s + 0.979·6-s + 0.822·7-s − 0.353·8-s + 0.919·9-s + 0.352·10-s − 0.930·11-s − 0.692·12-s + 0.676·13-s − 0.581·14-s + 0.691·15-s + 0.250·16-s + 1.18·17-s − 0.650·18-s − 1.08·19-s − 0.249·20-s − 1.13·21-s + 0.657·22-s − 0.785·23-s + 0.489·24-s − 0.751·25-s − 0.478·26-s + 0.111·27-s + 0.411·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 2.39T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 3.03T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 1.55T + 67T^{2} \) |
| 71 | \( 1 - 7.01T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 9.20T + 83T^{2} \) |
| 89 | \( 1 - 5.85T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.194204723091146079097518306789, −7.46863294700698415142006312085, −6.51925108991839034294650185518, −5.93269009335895467067559823702, −5.20769151481827045620304583914, −4.48808078088130808423560738497, −3.44653091034419765074445381161, −2.14361578619918769128470031449, −1.05527867445993174055077692482, 0,
1.05527867445993174055077692482, 2.14361578619918769128470031449, 3.44653091034419765074445381161, 4.48808078088130808423560738497, 5.20769151481827045620304583914, 5.93269009335895467067559823702, 6.51925108991839034294650185518, 7.46863294700698415142006312085, 8.194204723091146079097518306789