| L(s) = 1 | + i·5-s + 3.62·7-s + 6.20i·11-s − 0.578i·13-s − 1.42·17-s + 5.62i·19-s − 5.62·23-s − 25-s − 2i·29-s + 2.57·31-s + 3.62i·35-s − 7.83i·37-s − 5.25·41-s + 7.25i·43-s + 6.78·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.37·7-s + 1.87i·11-s − 0.160i·13-s − 0.344·17-s + 1.29i·19-s − 1.17·23-s − 0.200·25-s − 0.371i·29-s + 0.463·31-s + 0.613i·35-s − 1.28i·37-s − 0.820·41-s + 1.10i·43-s + 0.989·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.732607846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732607846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 - 6.20iT - 11T^{2} \) |
| 13 | \( 1 + 0.578iT - 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 7.83iT - 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 7.25iT - 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 2.20iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−9.938050853625027180367261150086, −8.880599719857831111025449584280, −7.80380776209041882169206740385, −7.60448807404857096268802131775, −6.50018058775710697729025678216, −5.52524643163758873239481048999, −4.57401080513214061480639821641, −3.95075975456067024044135275394, −2.33826258683604850619542317129, −1.64911576560271684594162071888,
0.70704721636664007484918652880, 1.96922259172711131695242183165, 3.24185733371795547997575237769, 4.37504795872916780924194513581, 5.13087514020431316779887899729, 5.93156286395819600450866071245, 6.90793712905777005865927382725, 8.047534672893356166980339519236, 8.468893330550148378611350667661, 9.068215689439634171243618224868
