| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 11-s + 4·13-s − 2·14-s + 16-s − 17-s − 5·19-s − 20-s − 22-s + 4·23-s + 25-s − 4·26-s + 2·28-s − 10·31-s − 32-s + 34-s − 2·35-s + 2·37-s + 5·38-s + 40-s − 41-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.14·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 1.79·31-s − 0.176·32-s + 0.171·34-s − 0.338·35-s + 0.328·37-s + 0.811·38-s + 0.158·40-s − 0.156·41-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364355462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364355462\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.14652259127150, −12.75014210791475, −12.22058465812743, −11.67725629634012, −11.10056775240913, −10.87190306086874, −10.71722989258296, −9.803553314770610, −9.309117558225382, −8.828761430118840, −8.522157409910351, −7.945135129592123, −7.636266484549667, −6.868966022206522, −6.587637615004924, −5.997287454335371, −5.328218521940203, −4.861151515592881, −4.061557075537019, −3.790106228761562, −3.071005083892020, −2.333530123711324, −1.692253122166038, −1.234731876536987, −0.3979716150923334,
0.3979716150923334, 1.234731876536987, 1.692253122166038, 2.333530123711324, 3.071005083892020, 3.790106228761562, 4.061557075537019, 4.861151515592881, 5.328218521940203, 5.997287454335371, 6.587637615004924, 6.868966022206522, 7.636266484549667, 7.945135129592123, 8.522157409910351, 8.828761430118840, 9.309117558225382, 9.803553314770610, 10.71722989258296, 10.87190306086874, 11.10056775240913, 11.67725629634012, 12.22058465812743, 12.75014210791475, 13.14652259127150
