| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 4·11-s − 13-s + 14-s + 16-s + 19-s + 20-s + 4·22-s + 4·23-s + 25-s − 26-s + 28-s + 5·29-s − 9·31-s + 32-s + 35-s + 4·37-s + 38-s + 40-s + 7·41-s − 43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.229·19-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 0.928·29-s − 1.61·31-s + 0.176·32-s + 0.169·35-s + 0.657·37-s + 0.162·38-s + 0.158·40-s + 1.09·41-s − 0.152·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.900680202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.900680202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.468222966172372372162048890535, −7.59391322935465543070602502966, −6.83236040340586528321107336584, −6.26603051379204416324743521148, −5.40431733074244560095699252545, −4.74366978263515871873275714602, −3.92331144475250149537089875303, −3.06968255027752379263750364261, −2.04858502766277408832903283922, −1.11641784642073869342281311014,
1.11641784642073869342281311014, 2.04858502766277408832903283922, 3.06968255027752379263750364261, 3.92331144475250149537089875303, 4.74366978263515871873275714602, 5.40431733074244560095699252545, 6.26603051379204416324743521148, 6.83236040340586528321107336584, 7.59391322935465543070602502966, 8.468222966172372372162048890535
