L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s − 5-s + 1.61·6-s − 2.23·8-s − 2·9-s − 1.61·10-s + 11-s + 0.618·12-s − 4.47·13-s − 15-s − 4.85·16-s + 4.23·17-s − 3.23·18-s + 1.47·19-s − 0.618·20-s + 1.61·22-s − 8.70·23-s − 2.23·24-s − 4·25-s − 7.23·26-s − 5·27-s + 4.47·29-s − 1.61·30-s − 0.236·31-s − 3.38·32-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.447·5-s + 0.660·6-s − 0.790·8-s − 0.666·9-s − 0.511·10-s + 0.301·11-s + 0.178·12-s − 1.24·13-s − 0.258·15-s − 1.21·16-s + 1.02·17-s − 0.762·18-s + 0.337·19-s − 0.138·20-s + 0.344·22-s − 1.81·23-s − 0.456·24-s − 0.800·25-s − 1.41·26-s − 0.962·27-s + 0.830·29-s − 0.295·30-s − 0.0423·31-s − 0.597·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 + 8.70T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 0.236T + 31T^{2} \) |
| 37 | \( 1 + 3.47T + 37T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 3.47T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 - 9.47T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−8.567918953136780633484257647889, −8.041921234879686164509206930548, −7.17627026258232825692998879268, −6.12306502516773626490928137484, −5.44690620456728997159752121778, −4.60921837020544138460621471676, −3.71531160719648925625651121301, −3.11039342214413083653250917454, −2.10972816409973249933079616325, 0,
2.10972816409973249933079616325, 3.11039342214413083653250917454, 3.71531160719648925625651121301, 4.60921837020544138460621471676, 5.44690620456728997159752121778, 6.12306502516773626490928137484, 7.17627026258232825692998879268, 8.041921234879686164509206930548, 8.567918953136780633484257647889