| L(s) = 1 | − 2-s + 0.354·3-s + 4-s − 1.85·5-s − 0.354·6-s + 4.73·7-s − 8-s − 2.87·9-s + 1.85·10-s − 1.61·11-s + 0.354·12-s − 6.52·13-s − 4.73·14-s − 0.658·15-s + 16-s − 0.103·17-s + 2.87·18-s − 6.71·19-s − 1.85·20-s + 1.67·21-s + 1.61·22-s + 5.93·23-s − 0.354·24-s − 1.54·25-s + 6.52·26-s − 2.08·27-s + 4.73·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.204·3-s + 0.5·4-s − 0.830·5-s − 0.144·6-s + 1.78·7-s − 0.353·8-s − 0.958·9-s + 0.587·10-s − 0.488·11-s + 0.102·12-s − 1.81·13-s − 1.26·14-s − 0.169·15-s + 0.250·16-s − 0.0252·17-s + 0.677·18-s − 1.53·19-s − 0.415·20-s + 0.366·21-s + 0.345·22-s + 1.23·23-s − 0.0723·24-s − 0.309·25-s + 1.28·26-s − 0.400·27-s + 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9387834185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9387834185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 0.354T + 3T^{2} \) |
| 5 | \( 1 + 1.85T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 6.52T + 13T^{2} \) |
| 17 | \( 1 + 0.103T + 17T^{2} \) |
| 19 | \( 1 + 6.71T + 19T^{2} \) |
| 23 | \( 1 - 5.93T + 23T^{2} \) |
| 29 | \( 1 + 3.82T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 - 0.538T + 41T^{2} \) |
| 43 | \( 1 - 7.38T + 43T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 + 6.29T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 6.42T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.311855393206952559392472652277, −7.79628969512147254303253060789, −7.45232497910267810591842596317, −6.41117476769236837965476744531, −5.22652915889369332611270835578, −4.84000924321770602185378479972, −3.86365355147390072780808847591, −2.53775088425322932628712975697, −2.13013679582558977668996463159, −0.58276049707713319003167109022,
0.58276049707713319003167109022, 2.13013679582558977668996463159, 2.53775088425322932628712975697, 3.86365355147390072780808847591, 4.84000924321770602185378479972, 5.22652915889369332611270835578, 6.41117476769236837965476744531, 7.45232497910267810591842596317, 7.79628969512147254303253060789, 8.311855393206952559392472652277
