L(s) = 1 | − 2.66·2-s + 0.744·3-s + 5.12·4-s − 2.89·5-s − 1.98·6-s − 0.268·7-s − 8.34·8-s − 2.44·9-s + 7.72·10-s + 5.35·11-s + 3.81·12-s + 3.85·13-s + 0.716·14-s − 2.15·15-s + 12.0·16-s − 3.41·17-s + 6.52·18-s − 4.28·19-s − 14.8·20-s − 0.199·21-s − 14.2·22-s − 23-s − 6.21·24-s + 3.38·25-s − 10.2·26-s − 4.05·27-s − 1.37·28-s + ⋯ |
L(s) = 1 | − 1.88·2-s + 0.429·3-s + 2.56·4-s − 1.29·5-s − 0.811·6-s − 0.101·7-s − 2.95·8-s − 0.815·9-s + 2.44·10-s + 1.61·11-s + 1.10·12-s + 1.06·13-s + 0.191·14-s − 0.556·15-s + 3.00·16-s − 0.827·17-s + 1.53·18-s − 0.983·19-s − 3.31·20-s − 0.0435·21-s − 3.04·22-s − 0.208·23-s − 1.26·24-s + 0.676·25-s − 2.01·26-s − 0.780·27-s − 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 0.744T + 3T^{2} \) |
| 5 | \( 1 + 2.89T + 5T^{2} \) |
| 7 | \( 1 + 0.268T + 7T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 2.62T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 6.27T + 73T^{2} \) |
| 79 | \( 1 + 9.39T + 79T^{2} \) |
| 83 | \( 1 - 4.93T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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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
−9.815006491926823479133408876182, −8.954262167316113198799726181593, −8.433941152479819533143481677310, −7.963637811416910341184872298305, −6.72921212006096165823081293229, −6.28731407433291691940558789655, −4.08350492745012857982159777156, −3.09372217530364009969402072205, −1.59468185351796211785457369986, 0,
1.59468185351796211785457369986, 3.09372217530364009969402072205, 4.08350492745012857982159777156, 6.28731407433291691940558789655, 6.72921212006096165823081293229, 7.963637811416910341184872298305, 8.433941152479819533143481677310, 8.954262167316113198799726181593, 9.815006491926823479133408876182