L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 5·11-s + 12-s − 14-s + 15-s + 16-s + 3·17-s + 18-s + 3·19-s + 20-s − 21-s + 5·22-s − 2·23-s + 24-s + 25-s + 27-s − 28-s + 3·29-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 + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.688·19-s + 0.223·20-s − 0.218·21-s + 1.06·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.721529700\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.721529700\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−14.80282683634466, −14.30876322460861, −14.01156564545816, −13.35365447315993, −13.06192448067886, −12.24808084648292, −11.92176179722954, −11.46101306101652, −10.69499845574523, −9.940627115389212, −9.708071438278064, −9.152204455102263, −8.404049163098924, −7.950089811244893, −7.136977807881607, −6.577650869711188, −6.326370160312184, −5.390104399680085, −5.015520038672589, −4.045892967356836, −3.654657929120107, −3.108638113580931, −2.295735426985873, −1.581519789825330, −0.8764581841918619,
0.8764581841918619, 1.581519789825330, 2.295735426985873, 3.108638113580931, 3.654657929120107, 4.045892967356836, 5.015520038672589, 5.390104399680085, 6.326370160312184, 6.577650869711188, 7.136977807881607, 7.950089811244893, 8.404049163098924, 9.152204455102263, 9.708071438278064, 9.940627115389212, 10.69499845574523, 11.46101306101652, 11.92176179722954, 12.24808084648292, 13.06192448067886, 13.35365447315993, 14.01156564545816, 14.30876322460861, 14.80282683634466