L(s) = 1 | − 1.86·3-s + (0.719 + 0.719i)7-s + 0.487·9-s + (0.805 − 0.805i)11-s + 5.90i·13-s + (−5.17 − 5.17i)17-s + (1.16 − 1.16i)19-s + (−1.34 − 1.34i)21-s + (−2.30 + 2.30i)23-s + 4.69·27-s + (3.71 + 3.71i)29-s − 9.82i·31-s + (−1.50 + 1.50i)33-s − 1.71i·37-s − 11.0i·39-s + ⋯ |
L(s) = 1 | − 1.07·3-s + (0.272 + 0.272i)7-s + 0.162·9-s + (0.242 − 0.242i)11-s + 1.63i·13-s + (−1.25 − 1.25i)17-s + (0.266 − 0.266i)19-s + (−0.293 − 0.293i)21-s + (−0.479 + 0.479i)23-s + 0.902·27-s + (0.690 + 0.690i)29-s − 1.76i·31-s + (−0.261 + 0.261i)33-s − 0.282i·37-s − 1.76i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1816479962\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1816479962\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.86T + 3T^{2} \) |
| 7 | \( 1 + (-0.719 - 0.719i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.805 + 0.805i)T - 11iT^{2} \) |
| 13 | \( 1 - 5.90iT - 13T^{2} \) |
| 17 | \( 1 + (5.17 + 5.17i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.16 + 1.16i)T - 19iT^{2} \) |
| 23 | \( 1 + (2.30 - 2.30i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.71 - 3.71i)T + 29iT^{2} \) |
| 31 | \( 1 + 9.82iT - 31T^{2} \) |
| 37 | \( 1 + 1.71iT - 37T^{2} \) |
| 41 | \( 1 - 3.93iT - 41T^{2} \) |
| 43 | \( 1 - 8.82iT - 43T^{2} \) |
| 47 | \( 1 + (3.21 - 3.21i)T - 47iT^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 + (5.24 + 5.24i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.59 + 1.59i)T - 61iT^{2} \) |
| 67 | \( 1 + 9.29iT - 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + (8.57 + 8.57i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 + (4.46 + 4.46i)T + 97iT^{2} \) |
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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
−9.245073368651269955515425634962, −8.365004881001289559553122100612, −7.25789319590445727904094847772, −6.51418726343729478613940572907, −5.94528453998604151922734710497, −4.81849373382811370863397391566, −4.41538181789696654921999947689, −2.90194208657964729898726744318, −1.67875631597815650040295554228, −0.085344692594744621277990933016,
1.30433355947111431654422180171, 2.77713923244397807026726434259, 4.01202396861309782724581281771, 4.89599754858759654650502573004, 5.70959477113871687196993140030, 6.35555359590095008987431050158, 7.19251629540225338172447705721, 8.257935904464203965659408676227, 8.724625127511671507661708857522, 10.16363472732164855078751469079