L(s) = 1 | + i·2-s − 3.14i·3-s − 4-s + (1.38 + 1.75i)5-s + 3.14·6-s − 3.50i·7-s − i·8-s − 6.86·9-s + (−1.75 + 1.38i)10-s + 2·11-s + 3.14i·12-s − 3.14i·13-s + 3.50·14-s + (5.50 − 4.36i)15-s + 16-s + i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.81i·3-s − 0.5·4-s + (0.621 + 0.783i)5-s + 1.28·6-s − 1.32i·7-s − 0.353i·8-s − 2.28·9-s + (−0.554 + 0.439i)10-s + 0.603·11-s + 0.906i·12-s − 0.871i·13-s + 0.936·14-s + (1.42 − 1.12i)15-s + 0.250·16-s + 0.242i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04755 - 0.506381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04755 - 0.506381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-1.38 - 1.75i)T \) |
| 17 | \( 1 - iT \) |
good | 3 | \( 1 + 3.14iT - 3T^{2} \) |
| 7 | \( 1 + 3.50iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.14iT - 13T^{2} \) |
| 19 | \( 1 - 3.86T + 19T^{2} \) |
| 23 | \( 1 - 7.50iT - 23T^{2} \) |
| 29 | \( 1 - 0.646T + 29T^{2} \) |
| 31 | \( 1 + 3.91T + 31T^{2} \) |
| 37 | \( 1 - 8.95iT - 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 - 4.28iT - 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 8.69iT - 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 + 0.910T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 + 3.14iT - 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 + 1.73iT - 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 - 2.41iT - 97T^{2} \) |
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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
−13.07658404310482880861235532353, −11.81903851695595316446881693894, −10.71674071170474310931172018317, −9.459567589321122516220892460424, −7.888810606401687350307816774916, −7.33753172193670558649994721704, −6.55730885033871885714749563519, −5.60140877042469304893742287704, −3.30899342325184736431873130847, −1.30711590399809800833614645236,
2.49878211766599260146951220993, 4.07353802233461188795951944979, 5.03779651211742696822438279832, 5.92852871120185376302129876859, 8.633607304282267166189177547652, 9.185770991845467508306027377481, 9.680971894598569644055726042676, 10.86973855501394886258207268557, 11.78527674172910679088883580801, 12.59571838914073293247337322614