| L(s) = 1 | + 2-s − 1.73·3-s + 4-s + 0.638·5-s − 1.73·6-s + 0.168·7-s + 8-s − 0.000964·9-s + 0.638·10-s + 0.464·11-s − 1.73·12-s − 2.40·13-s + 0.168·14-s − 1.10·15-s + 16-s + 5.16·17-s − 0.000964·18-s − 0.609·19-s + 0.638·20-s − 0.292·21-s + 0.464·22-s − 2.77·23-s − 1.73·24-s − 4.59·25-s − 2.40·26-s + 5.19·27-s + 0.168·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.999·3-s + 0.5·4-s + 0.285·5-s − 0.706·6-s + 0.0638·7-s + 0.353·8-s − 0.000321·9-s + 0.202·10-s + 0.140·11-s − 0.499·12-s − 0.666·13-s + 0.0451·14-s − 0.285·15-s + 0.250·16-s + 1.25·17-s − 0.000227·18-s − 0.139·19-s + 0.142·20-s − 0.0638·21-s + 0.0990·22-s − 0.577·23-s − 0.353·24-s − 0.918·25-s − 0.471·26-s + 1.00·27-s + 0.0319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.058997135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058997135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 - 0.638T + 5T^{2} \) |
| 7 | \( 1 - 0.168T + 7T^{2} \) |
| 11 | \( 1 - 0.464T + 11T^{2} \) |
| 13 | \( 1 + 2.40T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 + 0.609T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 - 2.99T + 29T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 - 7.99T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 + 6.88T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 + 2.57T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 4.91T + 97T^{2} \) |
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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
−8.235527473468718578861425682470, −7.58085842437607850234181685083, −6.71005784189375196296455431564, −6.02195776795338070947350993880, −5.52543079420603149334687468732, −4.85710511467626279978958816416, −4.01533887619004342761795281835, −3.02219899275236422200083375670, −2.05519290014748014292666370931, −0.77375700559169241812670461168,
0.77375700559169241812670461168, 2.05519290014748014292666370931, 3.02219899275236422200083375670, 4.01533887619004342761795281835, 4.85710511467626279978958816416, 5.52543079420603149334687468732, 6.02195776795338070947350993880, 6.71005784189375196296455431564, 7.58085842437607850234181685083, 8.235527473468718578861425682470
