L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 2·11-s + 12-s + 4·13-s + 14-s + 16-s + 18-s + 21-s + 2·22-s + 8·23-s + 24-s + 4·26-s + 27-s + 28-s + 4·31-s + 32-s + 2·33-s + 36-s − 2·37-s + 4·39-s − 6·41-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 + 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.426·22-s + 1.66·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.718·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.041403735\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.041403735\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.80719835473452, −12.26527949479244, −11.79782542794588, −11.33303505472530, −10.93410796016868, −10.55483935784957, −9.917357044889915, −9.443050913747678, −8.884882908856821, −8.474637663383799, −8.190782019604837, −7.445745598713163, −7.005408819717359, −6.601371651454347, −6.156259300817625, −5.395298656154335, −5.159144350232351, −4.405582024321593, −4.060693568188953, −3.493899505868956, −3.032314050352541, −2.521160937356467, −1.742252302060672, −1.301313691174573, −0.7028077255428153,
0.7028077255428153, 1.301313691174573, 1.742252302060672, 2.521160937356467, 3.032314050352541, 3.493899505868956, 4.060693568188953, 4.405582024321593, 5.159144350232351, 5.395298656154335, 6.156259300817625, 6.601371651454347, 7.005408819717359, 7.445745598713163, 8.190782019604837, 8.474637663383799, 8.884882908856821, 9.443050913747678, 9.917357044889915, 10.55483935784957, 10.93410796016868, 11.33303505472530, 11.79782542794588, 12.26527949479244, 12.80719835473452