| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 4·11-s − 12-s − 13-s − 14-s + 16-s + 17-s + 18-s + 4·19-s + 21-s − 4·22-s + 8·23-s − 24-s − 26-s − 27-s − 28-s + 2·29-s − 8·31-s + 32-s + 4·33-s + 34-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 − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.17421097541446, −12.69551024867424, −12.27865996923072, −11.90546070158911, −11.30037489690081, −10.79092708264778, −10.61382122476080, −10.02662097040330, −9.486526366430134, −9.022840712464805, −8.433821894106191, −7.643642040718000, −7.455363972156816, −6.885993461953966, −6.540485788488025, −5.631160018991992, −5.452515991199704, −5.131260473615614, −4.515306795893990, −3.840846215033406, −3.328516675156986, −2.765363418168176, −2.327937497961750, −1.447757998980346, −0.8100912233184165, 0,
0.8100912233184165, 1.447757998980346, 2.327937497961750, 2.765363418168176, 3.328516675156986, 3.840846215033406, 4.515306795893990, 5.131260473615614, 5.452515991199704, 5.631160018991992, 6.540485788488025, 6.885993461953966, 7.455363972156816, 7.643642040718000, 8.433821894106191, 9.022840712464805, 9.486526366430134, 10.02662097040330, 10.61382122476080, 10.79092708264778, 11.30037489690081, 11.90546070158911, 12.27865996923072, 12.69551024867424, 13.17421097541446
