L(s) = 1 | + 4.16·2-s + 3·3-s + 9.37·4-s + 18.6·5-s + 12.5·6-s − 32.7·7-s + 5.72·8-s + 9·9-s + 77.5·10-s + 11·11-s + 28.1·12-s − 88.0·13-s − 136.·14-s + 55.8·15-s − 51.1·16-s − 71.0·17-s + 37.5·18-s − 15.2·19-s + 174.·20-s − 98.2·21-s + 45.8·22-s − 103.·23-s + 17.1·24-s + 221.·25-s − 366.·26-s + 27·27-s − 307.·28-s + ⋯ |
L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.17·4-s + 1.66·5-s + 0.850·6-s − 1.76·7-s + 0.253·8-s + 0.333·9-s + 2.45·10-s + 0.301·11-s + 0.676·12-s − 1.87·13-s − 2.60·14-s + 0.961·15-s − 0.798·16-s − 1.01·17-s + 0.491·18-s − 0.183·19-s + 1.95·20-s − 1.02·21-s + 0.444·22-s − 0.938·23-s + 0.146·24-s + 1.77·25-s − 2.76·26-s + 0.192·27-s − 2.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
good | 2 | \( 1 - 4.16T + 8T^{2} \) |
| 5 | \( 1 - 18.6T + 125T^{2} \) |
| 7 | \( 1 + 32.7T + 343T^{2} \) |
| 13 | \( 1 + 88.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 177.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 369.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 399.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 88.2T + 2.05e5T^{2} \) |
| 67 | \( 1 + 596.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 804.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 498.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 358.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−8.629883220688277100338862848892, −7.03827735167383523179733574087, −6.54771803942598513137819384457, −6.09291844969471314806772813928, −5.08101332727873565972420517300, −4.41443775671141898571591635792, −3.20417154160281177346119094399, −2.65913428017054737101600457178, −1.98378645146303187545008981224, 0,
1.98378645146303187545008981224, 2.65913428017054737101600457178, 3.20417154160281177346119094399, 4.41443775671141898571591635792, 5.08101332727873565972420517300, 6.09291844969471314806772813928, 6.54771803942598513137819384457, 7.03827735167383523179733574087, 8.629883220688277100338862848892