L(s) = 1 | + (−1.75 + 2.21i)2-s + (−6.14 + 6.14i)3-s + (−1.85 − 7.78i)4-s + (−2.86 − 24.4i)6-s + (−16.4 − 16.4i)7-s + (20.5 + 9.53i)8-s − 48.5i·9-s + 44.6i·11-s + (59.2 + 36.4i)12-s + (0.849 + 0.849i)13-s + (65.3 − 7.66i)14-s + (−57.1 + 28.8i)16-s + (58.2 − 58.2i)17-s + (107. + 85.1i)18-s + 23.7·19-s + ⋯ |
L(s) = 1 | + (−0.619 + 0.784i)2-s + (−1.18 + 1.18i)3-s + (−0.231 − 0.972i)4-s + (−0.194 − 1.66i)6-s + (−0.888 − 0.888i)7-s + (0.906 + 0.421i)8-s − 1.79i·9-s + 1.22i·11-s + (1.42 + 0.877i)12-s + (0.0181 + 0.0181i)13-s + (1.24 − 0.146i)14-s + (−0.892 + 0.450i)16-s + (0.831 − 0.831i)17-s + (1.41 + 1.11i)18-s + 0.286·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.381771 - 0.0420315i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.381771 - 0.0420315i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.75 - 2.21i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (6.14 - 6.14i)T - 27iT^{2} \) |
| 7 | \( 1 + (16.4 + 16.4i)T + 343iT^{2} \) |
| 11 | \( 1 - 44.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-0.849 - 0.849i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-58.2 + 58.2i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 23.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-10.9 + 10.9i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + 127. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 253. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (92.9 - 92.9i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 98.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-235. + 235. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (250. + 250. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (149. + 149. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 12.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 332.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (199. + 199. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 664. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-699. - 699. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 703.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-940. + 940. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 386. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (1.06e3 - 1.06e3i)T - 9.12e5iT^{2} \) |
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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
−13.52974918889731387960015857892, −12.02401752261462034017890111860, −10.81366807075295332533947201601, −9.843900059869487973543774318525, −9.581130940596881406071485856090, −7.51206040043484782822595739296, −6.45752871180983572308484576834, −5.25057793188724101787173832504, −4.13254775665157609560696143187, −0.35761199967647738507165862711,
1.23117389372806789688863943585, 3.09429163953542792683786064899, 5.55660555171030835891551001578, 6.57670966102532641218522011581, 7.912770427685334679196114276547, 9.086100323517190636376826849116, 10.54327276366611288891299072369, 11.39332726034518285042714792180, 12.45021394056025240826607104947, 12.71942761709228480895241528234