| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 7·13-s + 16-s + 4·17-s − 18-s + 19-s + 22-s − 23-s + 24-s + 7·26-s − 27-s − 8·29-s + 6·31-s − 32-s + 33-s − 4·34-s + 36-s + 3·37-s − 38-s + 7·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.94·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 1.37·26-s − 0.192·27-s − 1.48·29-s + 1.07·31-s − 0.176·32-s + 0.174·33-s − 0.685·34-s + 1/6·36-s + 0.493·37-s − 0.162·38-s + 1.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−17.44928731148810, −16.77963262758127, −16.52697130679752, −15.79135025554324, −15.10353150005829, −14.62202894106123, −14.01665750611629, −13.01819689926425, −12.54623440680959, −11.90250925556790, −11.52283080269226, −10.64967535860848, −10.13576883032906, −9.617401549009076, −9.100581750463638, −8.037634029136000, −7.490820353257988, −7.178598498797547, −6.127434600604596, −5.559292555125702, −4.859066004863428, −4.012647805786473, −2.864623099255355, −2.214713427086070, −1.034465352669726, 0,
1.034465352669726, 2.214713427086070, 2.864623099255355, 4.012647805786473, 4.859066004863428, 5.559292555125702, 6.127434600604596, 7.178598498797547, 7.490820353257988, 8.037634029136000, 9.100581750463638, 9.617401549009076, 10.13576883032906, 10.64967535860848, 11.52283080269226, 11.90250925556790, 12.54623440680959, 13.01819689926425, 14.01665750611629, 14.62202894106123, 15.10353150005829, 15.79135025554324, 16.52697130679752, 16.77963262758127, 17.44928731148810
