| L(s) = 1 | − 1.67·2-s − 1.95·3-s + 0.807·4-s − 5-s + 3.27·6-s + 1.89·7-s + 1.99·8-s + 0.821·9-s + 1.67·10-s + 2.81·11-s − 1.57·12-s + 5.15·13-s − 3.17·14-s + 1.95·15-s − 4.96·16-s + 1.42·17-s − 1.37·18-s + 3.80·19-s − 0.807·20-s − 3.70·21-s − 4.72·22-s + 1.51·23-s − 3.90·24-s + 25-s − 8.64·26-s + 4.25·27-s + 1.52·28-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 1.12·3-s + 0.403·4-s − 0.447·5-s + 1.33·6-s + 0.716·7-s + 0.706·8-s + 0.273·9-s + 0.529·10-s + 0.849·11-s − 0.455·12-s + 1.43·13-s − 0.848·14-s + 0.504·15-s − 1.24·16-s + 0.346·17-s − 0.324·18-s + 0.872·19-s − 0.180·20-s − 0.808·21-s − 1.00·22-s + 0.316·23-s − 0.797·24-s + 0.200·25-s − 1.69·26-s + 0.819·27-s + 0.289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6909104349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6909104349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 1.42T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 - 8.97T + 29T^{2} \) |
| 31 | \( 1 - 7.19T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 7.08T + 41T^{2} \) |
| 43 | \( 1 + 9.89T + 43T^{2} \) |
| 47 | \( 1 + 3.86T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 + 9.70T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 6.13T + 83T^{2} \) |
| 89 | \( 1 - 3.18T + 89T^{2} \) |
| 97 | \( 1 + 5.46T + 97T^{2} \) |
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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
−8.904173839185860900346344571831, −8.544183917579654026352375465506, −7.78632850556637928496199077272, −6.82456951118611463821860457065, −6.21442524188419224278242865806, −5.11317807549963551182805491446, −4.46965558128787395778711339411, −3.29777691675711473069129766522, −1.44638334733063542428823340843, −0.811533784279997885530335855748,
0.811533784279997885530335855748, 1.44638334733063542428823340843, 3.29777691675711473069129766522, 4.46965558128787395778711339411, 5.11317807549963551182805491446, 6.21442524188419224278242865806, 6.82456951118611463821860457065, 7.78632850556637928496199077272, 8.544183917579654026352375465506, 8.904173839185860900346344571831
