| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 2·11-s + 4·13-s + 16-s + 4·17-s − 19-s + 2·20-s − 2·22-s − 8·23-s − 25-s − 4·26-s + 2·29-s + 2·31-s − 32-s − 4·34-s + 4·37-s + 38-s − 2·40-s − 6·41-s + 12·43-s + 2·44-s + 8·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.603·11-s + 1.10·13-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.447·20-s − 0.426·22-s − 1.66·23-s − 1/5·25-s − 0.784·26-s + 0.371·29-s + 0.359·31-s − 0.176·32-s − 0.685·34-s + 0.657·37-s + 0.162·38-s − 0.316·40-s − 0.937·41-s + 1.82·43-s + 0.301·44-s + 1.17·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.466713581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466713581\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.72226466937394, −14.20998276919975, −13.94764964136743, −13.31303345943763, −12.71677822126366, −12.07938506556417, −11.69581438463425, −11.05594565883789, −10.45444999777638, −9.960571427752283, −9.676314938516177, −8.872633815939959, −8.621621896635896, −7.805768537792895, −7.485768663652406, −6.517589560558937, −6.048800579293055, −5.890230213446997, −4.995824969972262, −4.035107916897455, −3.644980945822054, −2.689951350652200, −2.033287218058465, −1.379006933743311, −0.6873009286981165,
0.6873009286981165, 1.379006933743311, 2.033287218058465, 2.689951350652200, 3.644980945822054, 4.035107916897455, 4.995824969972262, 5.890230213446997, 6.048800579293055, 6.517589560558937, 7.485768663652406, 7.805768537792895, 8.621621896635896, 8.872633815939959, 9.676314938516177, 9.960571427752283, 10.45444999777638, 11.05594565883789, 11.69581438463425, 12.07938506556417, 12.71677822126366, 13.31303345943763, 13.94764964136743, 14.20998276919975, 14.72226466937394
