L(s) = 1 | − i·2-s − 4-s + 5-s + i·8-s − i·10-s − 1.53i·11-s + 1.48i·13-s + 16-s − 2.42·17-s − 4.86i·19-s − 20-s − 1.53·22-s − 0.267i·23-s + 25-s + 1.48·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447·5-s + 0.353i·8-s − 0.316i·10-s − 0.461i·11-s + 0.411i·13-s + 0.250·16-s − 0.589·17-s − 1.11i·19-s − 0.223·20-s − 0.326·22-s − 0.0558i·23-s + 0.200·25-s + 0.290·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9905210845\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9905210845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.53iT - 11T^{2} \) |
| 13 | \( 1 - 1.48iT - 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 + 4.86iT - 19T^{2} \) |
| 23 | \( 1 + 0.267iT - 23T^{2} \) |
| 29 | \( 1 + 0.898iT - 29T^{2} \) |
| 31 | \( 1 + 0.828iT - 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 41 | \( 1 + 8.76T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 - 3.47iT - 53T^{2} \) |
| 59 | \( 1 + 6.25T + 59T^{2} \) |
| 61 | \( 1 + 6.62iT - 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 0.343iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 14.9iT - 97T^{2} \) |
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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
−8.277092272554906134974072531353, −7.19673321368621034115216025122, −6.57260596859457581533476237454, −5.69070553828087357509416933867, −4.92399841377222002066511504385, −4.19062104554582808486983968169, −3.23396974342947320661707904740, −2.43223160127916858898359736240, −1.55252516738445416747450671253, −0.27434825705280080795486463784,
1.32259258796413273323949731623, 2.37894434976663706720384106985, 3.51421301937768711336184799320, 4.30799406560613346115428727016, 5.24748936680021062512816709428, 5.71757599721141004707251628183, 6.63588456047190169157130890201, 7.12388466018307750943200957516, 8.004410145478754028457061378342, 8.589686628356546280904213812343