L(s) = 1 | + 2-s + 1.41·3-s + 4-s + 5-s + 1.41·6-s − 3.15·7-s + 8-s − 1.01·9-s + 10-s − 1.04·11-s + 1.41·12-s − 3.01·13-s − 3.15·14-s + 1.41·15-s + 16-s + 2.09·17-s − 1.01·18-s + 3.54·19-s + 20-s − 4.44·21-s − 1.04·22-s − 1.80·23-s + 1.41·24-s + 25-s − 3.01·26-s − 5.65·27-s − 3.15·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.814·3-s + 0.5·4-s + 0.447·5-s + 0.575·6-s − 1.19·7-s + 0.353·8-s − 0.336·9-s + 0.316·10-s − 0.313·11-s + 0.407·12-s − 0.836·13-s − 0.842·14-s + 0.364·15-s + 0.250·16-s + 0.507·17-s − 0.238·18-s + 0.813·19-s + 0.223·20-s − 0.969·21-s − 0.221·22-s − 0.376·23-s + 0.287·24-s + 0.200·25-s − 0.591·26-s − 1.08·27-s − 0.595·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 + 0.429T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 + 7.43T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 7.88T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 + 4.36T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 8.28T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 8.34T + 89T^{2} \) |
| 97 | \( 1 - 6.35T + 97T^{2} \) |
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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
−7.73677989496445477167646254995, −6.91156070378793215582334031301, −6.30647614408021030717518153323, −5.46411481027013797980526141835, −4.99442146385039781559968509801, −3.69857585655747276547984671349, −3.26382834757902870553205956620, −2.60309295802346260743078151412, −1.73692362804224493196362655001, 0,
1.73692362804224493196362655001, 2.60309295802346260743078151412, 3.26382834757902870553205956620, 3.69857585655747276547984671349, 4.99442146385039781559968509801, 5.46411481027013797980526141835, 6.30647614408021030717518153323, 6.91156070378793215582334031301, 7.73677989496445477167646254995