| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s − 6·13-s − 2·15-s + 2·17-s − 8·19-s − 21-s − 4·23-s − 25-s + 27-s + 2·29-s − 8·31-s + 2·35-s − 6·37-s − 6·39-s + 2·41-s + 8·43-s − 2·45-s − 4·47-s + 49-s + 2·51-s − 2·53-s − 8·57-s + 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 1.83·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s − 0.298·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.43506547773995, −12.85988050449513, −12.60628543522407, −12.12409959418050, −11.78044566583290, −11.10394007030713, −10.56752633396718, −10.14819394942863, −9.683781100116631, −9.174388273202851, −8.656656300239655, −8.090086615552739, −7.808751428855513, −7.181264945941810, −6.888725025470485, −6.244187672178562, −5.533605154053738, −5.049820489886196, −4.309724668699712, −3.975012155812640, −3.529426013260171, −2.743206014542686, −2.238805944376964, −1.786588707866377, −0.5861229049956442, 0,
0.5861229049956442, 1.786588707866377, 2.238805944376964, 2.743206014542686, 3.529426013260171, 3.975012155812640, 4.309724668699712, 5.049820489886196, 5.533605154053738, 6.244187672178562, 6.888725025470485, 7.181264945941810, 7.808751428855513, 8.090086615552739, 8.656656300239655, 9.174388273202851, 9.683781100116631, 10.14819394942863, 10.56752633396718, 11.10394007030713, 11.78044566583290, 12.12409959418050, 12.60628543522407, 12.85988050449513, 13.43506547773995
