| L(s) = 1 | + 2-s − 3-s + 4-s + 0.815·5-s − 6-s + 2.48·7-s + 8-s + 9-s + 0.815·10-s + 5.73·11-s − 12-s + 5.67·13-s + 2.48·14-s − 0.815·15-s + 16-s + 6.55·17-s + 18-s + 4.56·19-s + 0.815·20-s − 2.48·21-s + 5.73·22-s − 23-s − 24-s − 4.33·25-s + 5.67·26-s − 27-s + 2.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.364·5-s − 0.408·6-s + 0.940·7-s + 0.353·8-s + 0.333·9-s + 0.257·10-s + 1.72·11-s − 0.288·12-s + 1.57·13-s + 0.665·14-s − 0.210·15-s + 0.250·16-s + 1.58·17-s + 0.235·18-s + 1.04·19-s + 0.182·20-s − 0.543·21-s + 1.22·22-s − 0.208·23-s − 0.204·24-s − 0.866·25-s + 1.11·26-s − 0.192·27-s + 0.470·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.975425724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.975425724\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.815T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 - 6.55T + 17T^{2} \) |
| 19 | \( 1 - 4.56T + 19T^{2} \) |
| 31 | \( 1 + 7.56T + 31T^{2} \) |
| 37 | \( 1 + 3.27T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 + 3.67T + 43T^{2} \) |
| 47 | \( 1 + 8.69T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 0.843T + 59T^{2} \) |
| 61 | \( 1 + 3.73T + 61T^{2} \) |
| 67 | \( 1 + 5.02T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 0.891T + 79T^{2} \) |
| 83 | \( 1 + 5.11T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−8.297166773629056185074129945014, −7.61823348487613467148529254020, −6.79538786767624958266698205441, −5.90345159934698278060592823301, −5.70025174782258736496924342026, −4.72130838849386305157055993617, −3.83121253362143127108976243104, −3.33719493697938944234572642628, −1.49882609782442780686443120990, −1.41274444555375300463519628703,
1.41274444555375300463519628703, 1.49882609782442780686443120990, 3.33719493697938944234572642628, 3.83121253362143127108976243104, 4.72130838849386305157055993617, 5.70025174782258736496924342026, 5.90345159934698278060592823301, 6.79538786767624958266698205441, 7.61823348487613467148529254020, 8.297166773629056185074129945014
