L(s) = 1 | + (0.0747 + 0.997i)4-s + (0.0546 + 0.139i)5-s + (0.365 + 0.930i)7-s + (0.955 + 0.294i)9-s + (−1.40 + 0.432i)11-s + (−0.988 + 0.149i)16-s + (−1.48 − 1.01i)17-s + (−0.5 + 0.866i)19-s + (−0.134 + 0.0648i)20-s + (1.57 − 1.07i)23-s + (0.716 − 0.664i)25-s + (−0.900 + 0.433i)28-s + (−0.109 + 0.101i)35-s + (−0.222 + 0.974i)36-s + (1.03 + 1.29i)43-s + (−0.535 − 1.36i)44-s + ⋯ |
L(s) = 1 | + (0.0747 + 0.997i)4-s + (0.0546 + 0.139i)5-s + (0.365 + 0.930i)7-s + (0.955 + 0.294i)9-s + (−1.40 + 0.432i)11-s + (−0.988 + 0.149i)16-s + (−1.48 − 1.01i)17-s + (−0.5 + 0.866i)19-s + (−0.134 + 0.0648i)20-s + (1.57 − 1.07i)23-s + (0.716 − 0.664i)25-s + (−0.900 + 0.433i)28-s + (−0.109 + 0.101i)35-s + (−0.222 + 0.974i)36-s + (1.03 + 1.29i)43-s + (−0.535 − 1.36i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.008430904\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.008430904\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.365 - 0.930i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 3 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 5 | \( 1 + (-0.0546 - 0.139i)T + (-0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \) |
| 23 | \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-1.44 - 1.34i)T + (0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 61 | \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.72174149266425635228336440325, −9.473854677991206266340143710479, −8.718245549403540652277460557908, −7.928176382077720567182558715310, −7.21341937229161831228118988772, −6.35204150823558704321042041821, −4.86734385978402774408406636654, −4.50567497067772240975691211077, −2.81174973857918583803341830656, −2.27404570499574768330633283635,
1.04169122193521846165461187052, 2.34799260557309966540538935776, 3.96144587941993442990663578473, 4.84848924788025128965622908315, 5.59857308213869688502557911956, 6.91195736322121498269771147290, 7.20173088554037290871365562346, 8.573180774915044983322099909071, 9.257681771873502785798344678269, 10.36578395269350878119750323101