| L(s) = 1 | − 2-s − 3-s + 4-s + 1.40·5-s + 6-s − 4.52·7-s − 8-s + 9-s − 1.40·10-s − 2.48·11-s − 12-s − 13-s + 4.52·14-s − 1.40·15-s + 16-s + 3.25·17-s − 18-s + 2.32·19-s + 1.40·20-s + 4.52·21-s + 2.48·22-s + 7.94·23-s + 24-s − 3.01·25-s + 26-s − 27-s − 4.52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.629·5-s + 0.408·6-s − 1.71·7-s − 0.353·8-s + 0.333·9-s − 0.445·10-s − 0.750·11-s − 0.288·12-s − 0.277·13-s + 1.21·14-s − 0.363·15-s + 0.250·16-s + 0.789·17-s − 0.235·18-s + 0.533·19-s + 0.314·20-s + 0.988·21-s + 0.530·22-s + 1.65·23-s + 0.204·24-s − 0.603·25-s + 0.196·26-s − 0.192·27-s − 0.855·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 + 6.13T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 - 9.38T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.80T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 - 9.79T + 61T^{2} \) |
| 67 | \( 1 + 1.67T + 67T^{2} \) |
| 71 | \( 1 - 8.11T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 + 2.03T + 83T^{2} \) |
| 89 | \( 1 - 8.03T + 89T^{2} \) |
| 97 | \( 1 - 9.30T + 97T^{2} \) |
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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
−7.41161716096671454170223139013, −6.70089726372143105162105225102, −6.31439613358719989410088699152, −5.45957919121600585793972612602, −5.03434984842994084691385685774, −3.58094535086141127944573581891, −3.06974228207160996658292820590, −2.16981748092747386188269128528, −0.986519549946495541230958889013, 0,
0.986519549946495541230958889013, 2.16981748092747386188269128528, 3.06974228207160996658292820590, 3.58094535086141127944573581891, 5.03434984842994084691385685774, 5.45957919121600585793972612602, 6.31439613358719989410088699152, 6.70089726372143105162105225102, 7.41161716096671454170223139013
