| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s − 2·13-s + 16-s − 17-s + 4·19-s − 20-s + 4·22-s + 25-s + 2·26-s + 2·29-s + 8·31-s − 32-s + 34-s + 6·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s − 4·44-s − 7·49-s − 50-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.392·26-s + 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s + 0.986·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s − 0.603·44-s − 49-s − 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9077872776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9077872776\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 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
−9.569434387346537777568385433984, −8.536260486993426222433680088038, −7.895479119611892740823321117532, −7.31854421695554503192670786651, −6.37327767973845489442596402709, −5.33994312375096426501337050826, −4.49484648317123978080474389404, −3.15571219135677541849378493447, −2.34322618202517557418539947506, −0.73132359746041352090488207698,
0.73132359746041352090488207698, 2.34322618202517557418539947506, 3.15571219135677541849378493447, 4.49484648317123978080474389404, 5.33994312375096426501337050826, 6.37327767973845489442596402709, 7.31854421695554503192670786651, 7.895479119611892740823321117532, 8.536260486993426222433680088038, 9.569434387346537777568385433984
