| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 11-s + 13-s − 14-s + 16-s + 17-s + 7·19-s + 20-s + 22-s − 3·23-s − 4·25-s + 26-s − 28-s + 3·29-s + 8·31-s + 32-s + 34-s − 35-s + 7·37-s + 7·38-s + 40-s − 8·41-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 + 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.60·19-s + 0.223·20-s + 0.213·22-s − 0.625·23-s − 4/5·25-s + 0.196·26-s − 0.188·28-s + 0.557·29-s + 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.169·35-s + 1.15·37-s + 1.13·38-s + 0.158·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.013799398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013799398\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.799161780228327520974515573275, −8.507140133505317474449494792816, −7.69488365555065663031941948688, −6.77874032891181696818904438957, −6.04012505199650201415683524492, −5.37271504020561549334482219473, −4.34865333294335971384701314535, −3.42544243473517943943741725271, −2.51773534345523556502922151340, −1.19342872714849413294161177850,
1.19342872714849413294161177850, 2.51773534345523556502922151340, 3.42544243473517943943741725271, 4.34865333294335971384701314535, 5.37271504020561549334482219473, 6.04012505199650201415683524492, 6.77874032891181696818904438957, 7.69488365555065663031941948688, 8.507140133505317474449494792816, 9.799161780228327520974515573275
