L(s) = 1 | + 2-s − 2.63·3-s + 4-s − 5-s − 2.63·6-s + 2.17·7-s + 8-s + 3.95·9-s − 10-s − 1.77·11-s − 2.63·12-s + 0.900·13-s + 2.17·14-s + 2.63·15-s + 16-s − 5.23·17-s + 3.95·18-s + 1.04·19-s − 20-s − 5.72·21-s − 1.77·22-s + 2.25·23-s − 2.63·24-s + 25-s + 0.900·26-s − 2.50·27-s + 2.17·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s − 0.447·5-s − 1.07·6-s + 0.820·7-s + 0.353·8-s + 1.31·9-s − 0.316·10-s − 0.535·11-s − 0.761·12-s + 0.249·13-s + 0.580·14-s + 0.680·15-s + 0.250·16-s − 1.26·17-s + 0.931·18-s + 0.239·19-s − 0.223·20-s − 1.24·21-s − 0.378·22-s + 0.469·23-s − 0.538·24-s + 0.200·25-s + 0.176·26-s − 0.482·27-s + 0.410·28-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 | \( 1 + 2.63T + 3T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 - 0.900T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 + 7.25T + 31T^{2} \) |
| 37 | \( 1 + 1.44T + 37T^{2} \) |
| 41 | \( 1 - 0.883T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 + 8.40T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 - 1.78T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 0.937T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−7.48894865764632906515010028775, −6.80182176902432635137217431813, −6.22545020015286694506167813176, −5.40787199516337193938441987380, −4.89322098556199430674884630320, −4.41831379461011145726942226150, −3.45628667377687588388914303503, −2.29059221173536670159801211739, −1.23231493508530474988739740789, 0,
1.23231493508530474988739740789, 2.29059221173536670159801211739, 3.45628667377687588388914303503, 4.41831379461011145726942226150, 4.89322098556199430674884630320, 5.40787199516337193938441987380, 6.22545020015286694506167813176, 6.80182176902432635137217431813, 7.48894865764632906515010028775