| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 11-s + 2·13-s + 15-s − 2·19-s − 2·21-s + 25-s + 27-s + 8·31-s − 33-s − 2·35-s − 2·37-s + 2·39-s − 2·43-s + 45-s − 3·49-s − 6·53-s − 55-s − 2·57-s − 12·59-s − 2·61-s − 2·63-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s − 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.43·31-s − 0.174·33-s − 0.338·35-s − 0.328·37-s + 0.320·39-s − 0.304·43-s + 0.149·45-s − 3/7·49-s − 0.824·53-s − 0.134·55-s − 0.264·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.88918069181021, −12.35297949575129, −12.06151410507118, −11.34376829138341, −10.86356467842573, −10.46976261080527, −9.962510440141801, −9.635764075569952, −9.125345551104643, −8.747013023180250, −8.166358383490996, −7.861107317507040, −7.262783207645058, −6.616577531345444, −6.282166401675739, −6.035260006088790, −5.122279498854078, −4.849452911090688, −4.178071975289860, −3.592961906409482, −3.153127643887820, −2.676282750514766, −2.089419856835267, −1.502642187344816, −0.8135433378666874, 0,
0.8135433378666874, 1.502642187344816, 2.089419856835267, 2.676282750514766, 3.153127643887820, 3.592961906409482, 4.178071975289860, 4.849452911090688, 5.122279498854078, 6.035260006088790, 6.282166401675739, 6.616577531345444, 7.262783207645058, 7.861107317507040, 8.166358383490996, 8.747013023180250, 9.125345551104643, 9.635764075569952, 9.962510440141801, 10.46976261080527, 10.86356467842573, 11.34376829138341, 12.06151410507118, 12.35297949575129, 12.88918069181021
