| L(s) = 1 | + 0.0683·2-s − 1.99·4-s + 2.66·5-s − 0.273·8-s + 0.182·10-s − 1.59·11-s − 5.25·13-s + 3.97·16-s + 6.54·17-s − 1.90·19-s − 5.31·20-s − 0.109·22-s − 3.06·23-s + 2.09·25-s − 0.359·26-s + 6.38·29-s − 6.71·31-s + 0.817·32-s + 0.447·34-s + 4.22·37-s − 0.130·38-s − 0.727·40-s − 7.39·41-s − 11.2·43-s + 3.19·44-s − 0.209·46-s − 3.79·47-s + ⋯ |
| L(s) = 1 | + 0.0483·2-s − 0.997·4-s + 1.19·5-s − 0.0965·8-s + 0.0575·10-s − 0.482·11-s − 1.45·13-s + 0.992·16-s + 1.58·17-s − 0.436·19-s − 1.18·20-s − 0.0233·22-s − 0.639·23-s + 0.419·25-s − 0.0704·26-s + 1.18·29-s − 1.20·31-s + 0.144·32-s + 0.0767·34-s + 0.695·37-s − 0.0210·38-s − 0.115·40-s − 1.15·41-s − 1.71·43-s + 0.481·44-s − 0.0309·46-s − 0.554·47-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 | \( 1 - 0.0683T + 2T^{2} \) |
| 5 | \( 1 - 2.66T + 5T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + 5.25T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 6.71T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 2.71T + 61T^{2} \) |
| 67 | \( 1 + 3.32T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 2.19T + 73T^{2} \) |
| 79 | \( 1 - 0.813T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 + 0.470T + 89T^{2} \) |
| 97 | \( 1 - 5.15T + 97T^{2} \) |
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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
−8.159694547711694687240213499911, −7.46659335732956365659419810745, −6.52385922980416507581712458901, −5.51736896844257446303899676083, −5.30769142401545492611076390014, −4.44109902112112586035309607973, −3.38000712280215978600265141996, −2.47594853371543896608838667043, −1.43731319965590403919249904623, 0,
1.43731319965590403919249904623, 2.47594853371543896608838667043, 3.38000712280215978600265141996, 4.44109902112112586035309607973, 5.30769142401545492611076390014, 5.51736896844257446303899676083, 6.52385922980416507581712458901, 7.46659335732956365659419810745, 8.159694547711694687240213499911
