| L(s) = 1 | + 2-s − 3-s + 4-s − 1.42·5-s − 6-s − 4.31·7-s + 8-s + 9-s − 1.42·10-s + 11-s − 12-s − 4.09·13-s − 4.31·14-s + 1.42·15-s + 16-s + 2.94·17-s + 18-s + 2.11·19-s − 1.42·20-s + 4.31·21-s + 22-s − 8.99·23-s − 24-s − 2.96·25-s − 4.09·26-s − 27-s − 4.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.637·5-s − 0.408·6-s − 1.63·7-s + 0.353·8-s + 0.333·9-s − 0.451·10-s + 0.301·11-s − 0.288·12-s − 1.13·13-s − 1.15·14-s + 0.368·15-s + 0.250·16-s + 0.715·17-s + 0.235·18-s + 0.484·19-s − 0.318·20-s + 0.941·21-s + 0.213·22-s − 1.87·23-s − 0.204·24-s − 0.593·25-s − 0.802·26-s − 0.192·27-s − 0.815·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.229449119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.229449119\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 + 8.99T + 23T^{2} \) |
| 29 | \( 1 + 0.364T + 29T^{2} \) |
| 31 | \( 1 - 6.51T + 31T^{2} \) |
| 37 | \( 1 - 8.86T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 9.40T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 0.972T + 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 2.11T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−8.168896436836542615626175375941, −7.54297309630527900193287183720, −6.79182080614083936069980788980, −6.18677354089540311868145547544, −5.57331547587401894783417962867, −4.60241423104244304136854819574, −3.84128805810088760924826044197, −3.21187924115478014913971282982, −2.20223192475796944584197385901, −0.55802501252205784211459793584,
0.55802501252205784211459793584, 2.20223192475796944584197385901, 3.21187924115478014913971282982, 3.84128805810088760924826044197, 4.60241423104244304136854819574, 5.57331547587401894783417962867, 6.18677354089540311868145547544, 6.79182080614083936069980788980, 7.54297309630527900193287183720, 8.168896436836542615626175375941
