| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s − 2·7-s − 8-s + 9-s + 4·10-s + 11-s + 12-s − 13-s + 2·14-s − 4·15-s + 16-s − 17-s − 18-s − 4·19-s − 4·20-s − 2·21-s − 22-s − 6·23-s − 24-s + 11·25-s + 26-s + 27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.894·20-s − 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 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
−16.25557131977370, −15.89100102873760, −15.32165057348103, −14.89555472386780, −14.42849754559807, −13.47950573987809, −12.93347820461488, −12.31650766493329, −11.74400315512770, −11.44329036870215, −10.60793604357791, −10.00181516008828, −9.531751868091784, −8.714663329380377, −8.154147057722409, −7.988233761959827, −7.160088345151468, −6.599167278785526, −6.085481112454584, −4.694984931567072, −4.252701369979912, −3.535438749395598, −2.931813966012393, −2.117263684890014, −0.8449661824052050, 0,
0.8449661824052050, 2.117263684890014, 2.931813966012393, 3.535438749395598, 4.252701369979912, 4.694984931567072, 6.085481112454584, 6.599167278785526, 7.160088345151468, 7.988233761959827, 8.154147057722409, 8.714663329380377, 9.531751868091784, 10.00181516008828, 10.60793604357791, 11.44329036870215, 11.74400315512770, 12.31650766493329, 12.93347820461488, 13.47950573987809, 14.42849754559807, 14.89555472386780, 15.32165057348103, 15.89100102873760, 16.25557131977370
