| L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s − 5·13-s + 15-s − 17-s + 8·19-s − 2·23-s − 4·25-s + 27-s + 9·29-s − 3·31-s + 4·33-s + 8·37-s − 5·39-s + 5·41-s − 4·43-s + 45-s + 9·47-s − 51-s − 12·53-s + 4·55-s + 8·57-s + 59-s − 6·61-s − 5·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 1.38·13-s + 0.258·15-s − 0.242·17-s + 1.83·19-s − 0.417·23-s − 4/5·25-s + 0.192·27-s + 1.67·29-s − 0.538·31-s + 0.696·33-s + 1.31·37-s − 0.800·39-s + 0.780·41-s − 0.609·43-s + 0.149·45-s + 1.31·47-s − 0.140·51-s − 1.64·53-s + 0.539·55-s + 1.05·57-s + 0.130·59-s − 0.768·61-s − 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.304898844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.304898844\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.53897743608798, −12.58419700897669, −12.48019190619928, −11.85525558760835, −11.51110574272894, −10.94951362530323, −10.16028467254117, −9.808832462636407, −9.458331995866370, −9.176763684454044, −8.479784036565376, −7.850208488488661, −7.528976513507794, −6.999323899582318, −6.452279597771982, −5.925752228507414, −5.374514223109379, −4.669277216590859, −4.356159783824356, −3.629310399806351, −3.042250188897964, −2.540243155509874, −1.919969064317601, −1.263596377232372, −0.6050850066858432,
0.6050850066858432, 1.263596377232372, 1.919969064317601, 2.540243155509874, 3.042250188897964, 3.629310399806351, 4.356159783824356, 4.669277216590859, 5.374514223109379, 5.925752228507414, 6.452279597771982, 6.999323899582318, 7.528976513507794, 7.850208488488661, 8.479784036565376, 9.176763684454044, 9.458331995866370, 9.808832462636407, 10.16028467254117, 10.94951362530323, 11.51110574272894, 11.85525558760835, 12.48019190619928, 12.58419700897669, 13.53897743608798
