| L(s) = 1 | − 2.18·2-s + 3·3-s − 3.20·4-s + 10.5·5-s − 6.56·6-s + 30.7·7-s + 24.5·8-s + 9·9-s − 23.0·10-s − 12.4·11-s − 9.62·12-s + 71.5·13-s − 67.2·14-s + 31.5·15-s − 28.0·16-s − 7.40·17-s − 19.7·18-s − 74.5·19-s − 33.7·20-s + 92.1·21-s + 27.3·22-s + 23·23-s + 73.6·24-s − 14.2·25-s − 156.·26-s + 27·27-s − 98.5·28-s + ⋯ |
| L(s) = 1 | − 0.773·2-s + 0.577·3-s − 0.401·4-s + 0.941·5-s − 0.446·6-s + 1.65·7-s + 1.08·8-s + 0.333·9-s − 0.728·10-s − 0.342·11-s − 0.231·12-s + 1.52·13-s − 1.28·14-s + 0.543·15-s − 0.438·16-s − 0.105·17-s − 0.257·18-s − 0.899·19-s − 0.377·20-s + 0.957·21-s + 0.265·22-s + 0.208·23-s + 0.626·24-s − 0.114·25-s − 1.18·26-s + 0.192·27-s − 0.665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.848028601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.848028601\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 2.18T + 8T^{2} \) |
| 5 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 - 30.7T + 343T^{2} \) |
| 11 | \( 1 + 12.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 71.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.40T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.5T + 6.85e3T^{2} \) |
| 31 | \( 1 - 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 416.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 48.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 251.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 590.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 442.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 95.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 84.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−8.803215954834187029724565219942, −8.071765940561916319681463056185, −7.80962871139704565872907503563, −6.47664736698470735229386411092, −5.60796836800675809321288085902, −4.64128756440984651385857833839, −4.04543434710117023096623314025, −2.50059143561440749637992154981, −1.62910018086611738495411257906, −0.957497924333345022333170551007,
0.957497924333345022333170551007, 1.62910018086611738495411257906, 2.50059143561440749637992154981, 4.04543434710117023096623314025, 4.64128756440984651385857833839, 5.60796836800675809321288085902, 6.47664736698470735229386411092, 7.80962871139704565872907503563, 8.071765940561916319681463056185, 8.803215954834187029724565219942
