L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 3.26·11-s − 12-s − 4.34·13-s + 14-s + 15-s + 16-s + 5.75·17-s + 18-s − 3.07·19-s − 20-s − 21-s + 3.26·22-s + 23-s − 24-s + 25-s − 4.34·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.983·11-s − 0.288·12-s − 1.20·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 1.39·17-s + 0.235·18-s − 0.706·19-s − 0.223·20-s − 0.218·21-s + 0.695·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s − 0.851·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.528817557\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.528817557\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 3.26T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 29 | \( 1 - 0.581T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 8.83T + 41T^{2} \) |
| 43 | \( 1 - 0.738T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 8.83T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 - 0.738T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 + 6.31T + 73T^{2} \) |
| 79 | \( 1 + 2.15T + 79T^{2} \) |
| 83 | \( 1 + 9.60T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 4.34T + 97T^{2} \) |
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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
−8.128253082664440785682982805402, −7.29253108572538689229701540718, −6.87202735818218635247875152603, −5.96852048835946297944909909176, −5.28914471373960708075962956732, −4.57947148124649940361072796556, −3.94513663367818623554297389662, −3.04177579686617557298860729394, −1.94886992326993975125391747074, −0.829292726497369960393528988290,
0.829292726497369960393528988290, 1.94886992326993975125391747074, 3.04177579686617557298860729394, 3.94513663367818623554297389662, 4.57947148124649940361072796556, 5.28914471373960708075962956732, 5.96852048835946297944909909176, 6.87202735818218635247875152603, 7.29253108572538689229701540718, 8.128253082664440785682982805402