| L(s) = 1 | − 0.495·2-s − 1.75·4-s − 3.69·5-s + 1.86·8-s + 1.83·10-s − 0.892·11-s − 1.19·13-s + 2.58·16-s + 0.249·17-s − 2.80·19-s + 6.47·20-s + 0.442·22-s + 2.47·23-s + 8.63·25-s + 0.593·26-s − 4.14·29-s + 3.58·31-s − 5.00·32-s − 0.123·34-s + 4.73·37-s + 1.39·38-s − 6.87·40-s + 4.78·41-s + 9.97·43-s + 1.56·44-s − 1.22·46-s − 10.1·47-s + ⋯ |
| L(s) = 1 | − 0.350·2-s − 0.877·4-s − 1.65·5-s + 0.658·8-s + 0.579·10-s − 0.269·11-s − 0.331·13-s + 0.646·16-s + 0.0606·17-s − 0.644·19-s + 1.44·20-s + 0.0943·22-s + 0.516·23-s + 1.72·25-s + 0.116·26-s − 0.769·29-s + 0.643·31-s − 0.884·32-s − 0.0212·34-s + 0.777·37-s + 0.225·38-s − 1.08·40-s + 0.746·41-s + 1.52·43-s + 0.236·44-s − 0.181·46-s − 1.48·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.495T + 2T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 + 0.892T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 0.249T + 17T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 9.88T + 53T^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 1.83T + 73T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.135564065691299098731523007365, −7.55913920392101028121914111752, −6.94850108131656899992240728954, −5.76138959401542372815544004861, −4.84708950233103732364006642132, −4.23072165103008672754690826145, −3.66162826145696150295508996356, −2.58937211709445121572507337245, −0.969219432270202720736370515596, 0,
0.969219432270202720736370515596, 2.58937211709445121572507337245, 3.66162826145696150295508996356, 4.23072165103008672754690826145, 4.84708950233103732364006642132, 5.76138959401542372815544004861, 6.94850108131656899992240728954, 7.55913920392101028121914111752, 8.135564065691299098731523007365
