| L(s) = 1 | − 2·2-s − 9.15·3-s + 4·4-s − 5·5-s + 18.3·6-s − 8·8-s + 56.7·9-s + 10·10-s − 35.7·11-s − 36.6·12-s − 67.4·13-s + 45.7·15-s + 16·16-s + 19.1·17-s − 113.·18-s + 86.6·19-s − 20·20-s + 71.5·22-s + 195.·23-s + 73.2·24-s + 25·25-s + 134.·26-s − 272.·27-s + 272.·29-s − 91.5·30-s − 21.6·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.447·5-s + 1.24·6-s − 0.353·8-s + 2.10·9-s + 0.316·10-s − 0.980·11-s − 0.880·12-s − 1.43·13-s + 0.787·15-s + 0.250·16-s + 0.273·17-s − 1.48·18-s + 1.04·19-s − 0.223·20-s + 0.693·22-s + 1.76·23-s + 0.622·24-s + 0.200·25-s + 1.01·26-s − 1.94·27-s + 1.74·29-s − 0.556·30-s − 0.125·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 9.15T + 27T^{2} \) |
| 11 | \( 1 + 35.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 195.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 67.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 107.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 609.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 834.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 491.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 479.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 784.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 227.T + 9.12e5T^{2} \) |
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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
−10.17431109945127628361340674812, −9.601534082076359268713126648636, −8.080925617067687651454174903338, −7.24717070653784102981482621915, −6.55678307611549608590222020445, −5.23270049071871663222005926102, −4.83413536585229832238433962156, −2.89492602483617122237615839473, −1.03549675197909689625523192878, 0,
1.03549675197909689625523192878, 2.89492602483617122237615839473, 4.83413536585229832238433962156, 5.23270049071871663222005926102, 6.55678307611549608590222020445, 7.24717070653784102981482621915, 8.080925617067687651454174903338, 9.601534082076359268713126648636, 10.17431109945127628361340674812
