L(s) = 1 | − 0.891·2-s − 3-s − 1.20·4-s + 2.60·5-s + 0.891·6-s − 3.10·7-s + 2.85·8-s + 9-s − 2.32·10-s − 1.58·11-s + 1.20·12-s + 2.28·13-s + 2.76·14-s − 2.60·15-s − 0.138·16-s + 17-s − 0.891·18-s − 0.262·19-s − 3.14·20-s + 3.10·21-s + 1.41·22-s − 1.11·23-s − 2.85·24-s + 1.81·25-s − 2.03·26-s − 27-s + 3.73·28-s + ⋯ |
L(s) = 1 | − 0.630·2-s − 0.577·3-s − 0.602·4-s + 1.16·5-s + 0.364·6-s − 1.17·7-s + 1.01·8-s + 0.333·9-s − 0.735·10-s − 0.476·11-s + 0.347·12-s + 0.633·13-s + 0.739·14-s − 0.673·15-s − 0.0347·16-s + 0.242·17-s − 0.210·18-s − 0.0601·19-s − 0.703·20-s + 0.677·21-s + 0.300·22-s − 0.232·23-s − 0.583·24-s + 0.362·25-s − 0.399·26-s − 0.192·27-s + 0.706·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 0.891T + 2T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 19 | \( 1 + 0.262T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 + 0.856T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 7.14T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 83 | \( 1 + 5.86T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−8.135536469222048286751713477283, −7.38796726080868461047501130974, −6.49537891094740429277660313743, −5.83815075083268420857216966137, −5.35774952388926286777729085383, −4.30209633089046749846715598699, −3.43032549205058393159858127651, −2.24643048164706829132924465054, −1.17096335059134473754709344184, 0,
1.17096335059134473754709344184, 2.24643048164706829132924465054, 3.43032549205058393159858127651, 4.30209633089046749846715598699, 5.35774952388926286777729085383, 5.83815075083268420857216966137, 6.49537891094740429277660313743, 7.38796726080868461047501130974, 8.135536469222048286751713477283