| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 2·15-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s + 4·22-s + 24-s − 25-s + 2·26-s + 27-s − 2·29-s − 2·30-s + 8·31-s + 32-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.49828547101072, −13.27543956809174, −12.55852730901803, −11.99093373162077, −11.85616118172244, −11.28733039918371, −10.79158589223650, −10.40269118581424, −9.498494168262533, −9.196372901836392, −8.779492036713733, −8.092782527355175, −7.754115358580081, −7.116117096997110, −6.758461689082947, −6.170145277917747, −5.727490540762138, −4.858724112282533, −4.326683570642734, −4.071362231694874, −3.530390286258680, −2.951525980134916, −2.387436857593290, −1.560508731339247, −1.056983237660951, 0,
1.056983237660951, 1.560508731339247, 2.387436857593290, 2.951525980134916, 3.530390286258680, 4.071362231694874, 4.326683570642734, 4.858724112282533, 5.727490540762138, 6.170145277917747, 6.758461689082947, 7.116117096997110, 7.754115358580081, 8.092782527355175, 8.779492036713733, 9.196372901836392, 9.498494168262533, 10.40269118581424, 10.79158589223650, 11.28733039918371, 11.85616118172244, 11.99093373162077, 12.55852730901803, 13.27543956809174, 13.49828547101072
