L(s) = 1 | + 2·2-s − 1.40·3-s + 4·4-s + 5·5-s − 2.80·6-s + 8·8-s − 25.0·9-s + 10·10-s + 0.0289·11-s − 5.60·12-s + 11.5·13-s − 7.00·15-s + 16·16-s + 42.8·17-s − 50.0·18-s + 158.·19-s + 20·20-s + 0.0579·22-s + 120.·23-s − 11.2·24-s + 25·25-s + 23.1·26-s + 72.8·27-s − 101.·29-s − 14.0·30-s + 74.1·31-s + 32·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.269·3-s + 0.5·4-s + 0.447·5-s − 0.190·6-s + 0.353·8-s − 0.927·9-s + 0.316·10-s + 0.000794·11-s − 0.134·12-s + 0.246·13-s − 0.120·15-s + 0.250·16-s + 0.610·17-s − 0.655·18-s + 1.91·19-s + 0.223·20-s + 0.000562·22-s + 1.09·23-s − 0.0953·24-s + 0.200·25-s + 0.174·26-s + 0.519·27-s − 0.647·29-s − 0.0852·30-s + 0.429·31-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
\(L(2)\) |
\(\approx\) |
\(3.078824238\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.078824238\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.40T + 27T^{2} \) |
| 11 | \( 1 - 0.0289T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 74.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 91.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 92.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 298.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 670.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 95.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 387.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 479.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 617.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 562.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 8.95T + 4.93e5T^{2} \) |
| 83 | \( 1 - 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74169457364008417357620469653, −9.751886910191911418652387291651, −8.803634897768400251700987255467, −7.64910422945832878001170548633, −6.68893044344696911925032776665, −5.56512197265694080838794436137, −5.18855005854684976111013779027, −3.60556802377618300984388321710, −2.67095951472011514260302471967, −1.05757733342284150285822267512,
1.05757733342284150285822267512, 2.67095951472011514260302471967, 3.60556802377618300984388321710, 5.18855005854684976111013779027, 5.56512197265694080838794436137, 6.68893044344696911925032776665, 7.64910422945832878001170548633, 8.803634897768400251700987255467, 9.751886910191911418652387291651, 10.74169457364008417357620469653