L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s + 12-s + 2·13-s + 2·15-s + 16-s − 3·17-s + 18-s + 7·19-s + 2·20-s + 7·23-s + 24-s − 25-s + 2·26-s + 27-s + 3·29-s + 2·30-s + 32-s − 3·34-s + 36-s − 7·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.60·19-s + 0.447·20-s + 1.45·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.557·29-s + 0.365·30-s + 0.176·32-s − 0.514·34-s + 1/6·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.187972681\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.187972681\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.82683423160294, −14.20006646160010, −13.90214557613579, −13.38671848039319, −13.09612531016536, −12.48309837796473, −11.77323517276610, −11.32370107249327, −10.72204402844748, −10.08550996100038, −9.663701842431350, −8.965723700875859, −8.634904457355266, −7.809344394087807, −7.170340400349906, −6.739398513926348, −6.091840464176207, −5.409478659777184, −5.020284492924170, −4.282768133266577, −3.461267990443810, −3.054950644611646, −2.308509639701325, −1.629459948249312, −0.8940489523360479,
0.8940489523360479, 1.629459948249312, 2.308509639701325, 3.054950644611646, 3.461267990443810, 4.282768133266577, 5.020284492924170, 5.409478659777184, 6.091840464176207, 6.739398513926348, 7.170340400349906, 7.809344394087807, 8.634904457355266, 8.965723700875859, 9.663701842431350, 10.08550996100038, 10.72204402844748, 11.32370107249327, 11.77323517276610, 12.48309837796473, 13.09612531016536, 13.38671848039319, 13.90214557613579, 14.20006646160010, 14.82683423160294