L(s) = 1 | + (−0.0475 + 0.998i)3-s + (0.327 + 0.945i)4-s + (0.415 − 0.909i)7-s + (−0.995 − 0.0950i)9-s + (−0.959 + 0.281i)12-s + (−0.195 + 0.807i)13-s + (−0.786 + 0.618i)16-s + (0.189 + 1.98i)19-s + (0.888 + 0.458i)21-s + (0.723 + 0.690i)25-s + (0.142 − 0.989i)27-s + (0.995 + 0.0950i)28-s + (−0.205 + 0.196i)31-s + (−0.235 − 0.971i)36-s + (−0.723 − 1.25i)37-s + ⋯ |
L(s) = 1 | + (−0.0475 + 0.998i)3-s + (0.327 + 0.945i)4-s + (0.415 − 0.909i)7-s + (−0.995 − 0.0950i)9-s + (−0.959 + 0.281i)12-s + (−0.195 + 0.807i)13-s + (−0.786 + 0.618i)16-s + (0.189 + 1.98i)19-s + (0.888 + 0.458i)21-s + (0.723 + 0.690i)25-s + (0.142 − 0.989i)27-s + (0.995 + 0.0950i)28-s + (−0.205 + 0.196i)31-s + (−0.235 − 0.971i)36-s + (−0.723 − 1.25i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.106568769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.106568769\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.0475 - 0.998i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (-0.959 + 0.281i)T \) |
good | 2 | \( 1 + (-0.327 - 0.945i)T^{2} \) |
| 5 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 11 | \( 1 + (0.723 + 0.690i)T^{2} \) |
| 13 | \( 1 + (0.195 - 0.807i)T + (-0.888 - 0.458i)T^{2} \) |
| 17 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.189 - 1.98i)T + (-0.981 + 0.189i)T^{2} \) |
| 23 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.205 - 0.196i)T + (0.0475 - 0.998i)T^{2} \) |
| 37 | \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 43 | \( 1 + (-0.425 + 0.368i)T + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 53 | \( 1 + (0.928 + 0.371i)T^{2} \) |
| 59 | \( 1 + (-0.888 + 0.458i)T^{2} \) |
| 61 | \( 1 + (-0.0883 + 0.0353i)T + (0.723 - 0.690i)T^{2} \) |
| 71 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 73 | \( 1 + (-1.22 - 0.175i)T + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (0.388 + 1.32i)T + (-0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (0.723 + 0.690i)T^{2} \) |
| 89 | \( 1 + (-0.995 + 0.0950i)T^{2} \) |
| 97 | \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.07540418610243834328391294036, −9.173368121616480003720165141751, −8.413961305900817678323173881539, −7.63491192759919844553051351414, −6.90293754222054285939379048791, −5.79567320094437214225563744683, −4.73730989523265134215672634605, −3.89119462172122166592065521473, −3.39399157638198036425214688226, −1.93333754539138299617388778221,
0.948077764025661497540929577030, 2.26059645809018018219264242529, 2.88082074858647685223307476729, 4.90661875802464345734879396717, 5.34778348239569976926981530623, 6.34011715572310693632550363434, 6.89103077668569545862340335219, 7.86582464403587965879408527391, 8.700554066413266512734868289357, 9.366399026780395193693693815058