L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 3·11-s − 12-s − 4·13-s − 14-s + 16-s + 2·17-s − 18-s − 21-s + 3·22-s + 8·23-s + 24-s + 4·26-s − 27-s + 28-s − 2·29-s − 5·31-s − 32-s + 3·33-s − 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.218·21-s + 0.639·22-s + 1.66·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.898·31-s − 0.176·32-s + 0.522·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5189201263\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5189201263\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.38152771910945, −12.02337814149526, −11.34194953024583, −11.18374975710492, −10.62463950847952, −10.26766257015262, −9.845760379884020, −9.341317647141724, −8.815360253284473, −8.527504581948631, −7.635484190360597, −7.495616345548002, −7.201876515261426, −6.605159993544820, −5.894598508521217, −5.466048655379957, −5.165685641394637, −4.585108408403356, −4.051977128759567, −3.246596835006109, −2.728735155529404, −2.313665694597505, −1.527932462496335, −1.046544609959067, −0.2384607399515862,
0.2384607399515862, 1.046544609959067, 1.527932462496335, 2.313665694597505, 2.728735155529404, 3.246596835006109, 4.051977128759567, 4.585108408403356, 5.165685641394637, 5.466048655379957, 5.894598508521217, 6.605159993544820, 7.201876515261426, 7.495616345548002, 7.635484190360597, 8.527504581948631, 8.815360253284473, 9.341317647141724, 9.845760379884020, 10.26766257015262, 10.62463950847952, 11.18374975710492, 11.34194953024583, 12.02337814149526, 12.38152771910945