L(s) = 1 | + 5i·2-s − 3i·3-s − 17·4-s + 15·6-s + 3i·7-s − 45i·8-s − 9·9-s − 11·11-s + 51i·12-s − 32i·13-s − 15·14-s + 89·16-s + 33i·17-s − 45i·18-s − 47·19-s + ⋯ |
L(s) = 1 | + 1.76i·2-s − 0.577i·3-s − 2.12·4-s + 1.02·6-s + 0.161i·7-s − 1.98i·8-s − 0.333·9-s − 0.301·11-s + 1.22i·12-s − 0.682i·13-s − 0.286·14-s + 1.39·16-s + 0.470i·17-s − 0.589i·18-s − 0.567·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.410612480\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.410612480\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 5iT - 8T^{2} \) |
| 7 | \( 1 - 3iT - 343T^{2} \) |
| 13 | \( 1 + 32iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 33iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 47T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 54T + 2.43e4T^{2} \) |
| 31 | \( 1 - 178T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 139T + 6.89e4T^{2} \) |
| 43 | \( 1 - 308iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 195iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 152iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 625T + 2.05e5T^{2} \) |
| 61 | \( 1 - 320T + 2.26e5T^{2} \) |
| 67 | \( 1 - 200iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 947T + 3.57e5T^{2} \) |
| 73 | \( 1 - 448iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 721T + 4.93e5T^{2} \) |
| 83 | \( 1 + 142iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 404T + 7.04e5T^{2} \) |
| 97 | \( 1 - 79iT - 9.12e5T^{2} \) |
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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
−9.842417329751743445817387824533, −8.699171779132760610936321761552, −8.252021314928925186019479405980, −7.51150433612711306487467253938, −6.58501267153422060534714171772, −6.02899905503742574519540327488, −5.12592780607657543352540847346, −4.16645588046454877189034600445, −2.60213957766508378798122485212, −0.73231626125972804485249840533,
0.58959185299038811259206839098, 1.94195649750244010518997734865, 2.91973842983730435902445869112, 3.91584705611531397102102537421, 4.60620891709721076657122269766, 5.61325758349549965541769990694, 7.04892339545878997809637865298, 8.360905321155435004219060395628, 9.090754077471905503795104381851, 9.841139287661426710788325195885