| L(s) = 1 | + 0.330·2-s + 5.27·3-s − 7.89·4-s − 9.14·5-s + 1.74·6-s + 25.7·7-s − 5.25·8-s + 0.787·9-s − 3.02·10-s − 46.9·11-s − 41.5·12-s + 65.6·13-s + 8.51·14-s − 48.1·15-s + 61.3·16-s + 6.50·17-s + 0.260·18-s − 20.7·19-s + 72.1·20-s + 135.·21-s − 15.5·22-s − 23·23-s − 27.7·24-s − 41.4·25-s + 21.7·26-s − 138.·27-s − 203.·28-s + ⋯ |
| L(s) = 1 | + 0.116·2-s + 1.01·3-s − 0.986·4-s − 0.817·5-s + 0.118·6-s + 1.38·7-s − 0.232·8-s + 0.0291·9-s − 0.0956·10-s − 1.28·11-s − 1.00·12-s + 1.40·13-s + 0.162·14-s − 0.829·15-s + 0.959·16-s + 0.0928·17-s + 0.00341·18-s − 0.250·19-s + 0.806·20-s + 1.40·21-s − 0.150·22-s − 0.208·23-s − 0.235·24-s − 0.331·25-s + 0.163·26-s − 0.984·27-s − 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 0.330T + 8T^{2} \) |
| 3 | \( 1 - 5.27T + 27T^{2} \) |
| 5 | \( 1 + 9.14T + 125T^{2} \) |
| 7 | \( 1 - 25.7T + 343T^{2} \) |
| 11 | \( 1 + 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.50T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.7T + 6.85e3T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 163.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 419.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 431.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 672.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 453.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 958.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 120.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 946.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 144.T + 9.12e5T^{2} \) |
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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
−9.398477244939749071522454036039, −8.518802422046750599032199907448, −8.121381477245612165593613260029, −7.63382957060420030561447404664, −5.81453751421035858869125159871, −4.88489026191125510240224868466, −3.97913064934423899747948710041, −3.13544278091014133579362590713, −1.65306676486689623978103047153, 0,
1.65306676486689623978103047153, 3.13544278091014133579362590713, 3.97913064934423899747948710041, 4.88489026191125510240224868466, 5.81453751421035858869125159871, 7.63382957060420030561447404664, 8.121381477245612165593613260029, 8.518802422046750599032199907448, 9.398477244939749071522454036039
