L(s) = 1 | + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (0.698 + 0.449i)7-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)12-s + (−0.239 − 1.66i)13-s + (−0.654 + 0.755i)16-s + (−1.61 + 1.03i)19-s + (−0.544 − 0.627i)21-s + (−0.142 − 0.989i)25-s + (−0.654 − 0.755i)27-s + (−0.118 + 0.822i)28-s + (0.186 − 1.29i)31-s + (−0.142 + 0.989i)36-s − 0.284·37-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)3-s + (0.415 + 0.909i)4-s + (0.698 + 0.449i)7-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)12-s + (−0.239 − 1.66i)13-s + (−0.654 + 0.755i)16-s + (−1.61 + 1.03i)19-s + (−0.544 − 0.627i)21-s + (−0.142 − 0.989i)25-s + (−0.654 − 0.755i)27-s + (−0.118 + 0.822i)28-s + (0.186 − 1.29i)31-s + (−0.142 + 0.989i)36-s − 0.284·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6058731517\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6058731517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
good | 2 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 7 | \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 11 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 13 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 23 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + 0.284T + T^{2} \) |
| 41 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 43 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 53 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 89 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + 1.91T + T^{2} \) |
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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
−12.55640639689542369176864540021, −11.91060328658130514310492137473, −10.95625719586983058202825656582, −10.16714868730782212709426247814, −8.310512894255061027012426427526, −7.84053137072750613844789626342, −6.50757408678597100604034450616, −5.50950457490169342739397703299, −4.14287885964806557761753126274, −2.29956614588298711166077894075,
1.74606183655345839181257615387, 4.33583321075802183371986404210, 5.13560550894647523254990130729, 6.50042205016937012463619097690, 7.07038311488843842856600957351, 8.900441216588631705890431366014, 9.904453240060501194748925808107, 10.99765077691192919185943307672, 11.24883087285052313886764350612, 12.36837185488386301143517757923