L(s) = 1 | − 2.24·2-s + 0.297·3-s + 3.02·4-s + 1.49·5-s − 0.667·6-s + 2.00·7-s − 2.29·8-s − 2.91·9-s − 3.35·10-s − 0.727·11-s + 0.901·12-s − 6.17·13-s − 4.49·14-s + 0.446·15-s − 0.899·16-s − 17-s + 6.52·18-s − 3.49·19-s + 4.53·20-s + 0.596·21-s + 1.63·22-s − 5.48·23-s − 0.684·24-s − 2.75·25-s + 13.8·26-s − 1.76·27-s + 6.06·28-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.171·3-s + 1.51·4-s + 0.669·5-s − 0.272·6-s + 0.757·7-s − 0.812·8-s − 0.970·9-s − 1.06·10-s − 0.219·11-s + 0.260·12-s − 1.71·13-s − 1.20·14-s + 0.115·15-s − 0.224·16-s − 0.242·17-s + 1.53·18-s − 0.801·19-s + 1.01·20-s + 0.130·21-s + 0.347·22-s − 1.14·23-s − 0.139·24-s − 0.551·25-s + 2.71·26-s − 0.338·27-s + 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 - 0.297T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 - 2.00T + 7T^{2} \) |
| 11 | \( 1 + 0.727T + 11T^{2} \) |
| 13 | \( 1 + 6.17T + 13T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 - 7.44T + 29T^{2} \) |
| 31 | \( 1 + 4.37T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 7.24T + 41T^{2} \) |
| 47 | \( 1 + 0.886T + 47T^{2} \) |
| 53 | \( 1 - 6.71T + 53T^{2} \) |
| 59 | \( 1 - 7.08T + 59T^{2} \) |
| 61 | \( 1 - 1.56T + 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 2.66T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 2.32T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 + 1.30T + 97T^{2} \) |
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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
−9.986426735464371452212739852933, −9.063420295020176110904970717093, −8.330148510885956652729408828575, −7.73649119909683128436427399903, −6.74405165922723822275738692125, −5.66088088436561406179123231277, −4.56610985216660071580919635244, −2.58519782493905735782650059119, −1.89555902249478005299047405181, 0,
1.89555902249478005299047405181, 2.58519782493905735782650059119, 4.56610985216660071580919635244, 5.66088088436561406179123231277, 6.74405165922723822275738692125, 7.73649119909683128436427399903, 8.330148510885956652729408828575, 9.063420295020176110904970717093, 9.986426735464371452212739852933