L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 4·11-s + 12-s − 14-s − 15-s − 16-s − 17-s − 18-s + 8·19-s − 20-s − 21-s − 4·22-s + 4·23-s − 3·24-s + 25-s − 27-s − 28-s − 2·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.218·21-s − 0.852·22-s + 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.95322936850142, −12.35922629934465, −11.85438089508709, −11.52363193835046, −11.03029894949319, −10.62135431939270, −10.00488323918902, −9.692100395315742, −9.256829815055946, −8.868869911657240, −8.483877724792652, −7.720601807746588, −7.412333816092763, −6.848890315235495, −6.528281885735371, −5.693100515818660, −5.278939356638835, −5.035504156140172, −4.341291176935293, −3.753996696283702, −3.409736578289066, −2.456825568112116, −1.776032549969245, −1.147471813491117, −0.9500353646224634, 0,
0.9500353646224634, 1.147471813491117, 1.776032549969245, 2.456825568112116, 3.409736578289066, 3.753996696283702, 4.341291176935293, 5.035504156140172, 5.278939356638835, 5.693100515818660, 6.528281885735371, 6.848890315235495, 7.412333816092763, 7.720601807746588, 8.483877724792652, 8.868869911657240, 9.256829815055946, 9.692100395315742, 10.00488323918902, 10.62135431939270, 11.03029894949319, 11.52363193835046, 11.85438089508709, 12.35922629934465, 12.95322936850142