L(s) = 1 | − i·2-s + 2.82·3-s − 4-s + (0.707 − 2.12i)5-s − 2.82i·6-s − 4.24·7-s + i·8-s + 5.00·9-s + (−2.12 − 0.707i)10-s − 2.82·12-s + 6i·13-s + 4.24i·14-s + (2.00 − 6i)15-s + 16-s + (2.82 + 3i)17-s − 5.00i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.63·3-s − 0.5·4-s + (0.316 − 0.948i)5-s − 1.15i·6-s − 1.60·7-s + 0.353i·8-s + 1.66·9-s + (−0.670 − 0.223i)10-s − 0.816·12-s + 1.66i·13-s + 1.13i·14-s + (0.516 − 1.54i)15-s + 0.250·16-s + (0.685 + 0.727i)17-s − 1.17i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36552 - 0.871965i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36552 - 0.871965i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 2.12i)T \) |
| 17 | \( 1 + (-2.82 - 3i)T \) |
good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 4.24iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 4.24iT - 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 4.24iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−12.76340090201164043951785380031, −11.95466595037747718627858757914, −10.08505581735487839464065369489, −9.418839338644338202652010668407, −8.956520327368728494365539279328, −7.77265927397674281762515943603, −6.23266659380700618461353091116, −4.28697831313215297950563479453, −3.31581171376736480719605291617, −1.91505430413316352158022021370,
2.97024208037910435614385931099, 3.40772590844707343908426910388, 5.66876859062903956842955748051, 6.95699127394199851292202224631, 7.65822435266637210201028614554, 8.860554645954631556466625401263, 9.804568842318255675860229131819, 10.33506300331523678719266673261, 12.49374556968304931490314759437, 13.30956120658930436056588348478