| L(s) = 1 | − 22·5-s + 7·7-s + 26·11-s − 54·13-s − 74·17-s − 116·19-s − 58·23-s + 359·25-s − 208·29-s + 252·31-s − 154·35-s + 50·37-s − 126·41-s − 164·43-s − 444·47-s + 49·49-s − 12·53-s − 572·55-s + 124·59-s − 162·61-s + 1.18e3·65-s + 860·67-s − 238·71-s − 146·73-s + 182·77-s + 984·79-s + 656·83-s + ⋯ |
| L(s) = 1 | − 1.96·5-s + 0.377·7-s + 0.712·11-s − 1.15·13-s − 1.05·17-s − 1.40·19-s − 0.525·23-s + 2.87·25-s − 1.33·29-s + 1.46·31-s − 0.743·35-s + 0.222·37-s − 0.479·41-s − 0.581·43-s − 1.37·47-s + 1/7·49-s − 0.0311·53-s − 1.40·55-s + 0.273·59-s − 0.340·61-s + 2.26·65-s + 1.56·67-s − 0.397·71-s − 0.234·73-s + 0.269·77-s + 1.40·79-s + 0.867·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6968251265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6968251265\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 208 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 - 50 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 444 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 T + p^{3} T^{2} \) |
| 59 | \( 1 - 124 T + p^{3} T^{2} \) |
| 61 | \( 1 + 162 T + p^{3} T^{2} \) |
| 67 | \( 1 - 860 T + p^{3} T^{2} \) |
| 71 | \( 1 + 238 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 984 T + p^{3} T^{2} \) |
| 83 | \( 1 - 656 T + p^{3} T^{2} \) |
| 89 | \( 1 - 954 T + p^{3} T^{2} \) |
| 97 | \( 1 - 526 T + p^{3} T^{2} \) |
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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
−9.481679525914374118451559146435, −8.475402570734784235831987313834, −8.041884083837483575515424782711, −7.11483774339827216498794007421, −6.46829760684640764206308563866, −4.82190599020899589324904505946, −4.33574878573133982038182793904, −3.46371767964374454963048050045, −2.12127560338025630936786181286, −0.42636953847691539119698838993,
0.42636953847691539119698838993, 2.12127560338025630936786181286, 3.46371767964374454963048050045, 4.33574878573133982038182793904, 4.82190599020899589324904505946, 6.46829760684640764206308563866, 7.11483774339827216498794007421, 8.041884083837483575515424782711, 8.475402570734784235831987313834, 9.481679525914374118451559146435
