L(s) = 1 | + 2·2-s − 1.44·3-s + 4·4-s − 19.6·5-s − 2.89·6-s + 11.5·7-s + 8·8-s − 24.8·9-s − 39.3·10-s + 37.6·11-s − 5.79·12-s − 44.5·13-s + 23.1·14-s + 28.5·15-s + 16·16-s + 61.1·17-s − 49.7·18-s − 63.7·19-s − 78.7·20-s − 16.8·21-s + 75.3·22-s + 177.·23-s − 11.5·24-s + 262.·25-s − 89.1·26-s + 75.2·27-s + 46.3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.278·3-s + 0.5·4-s − 1.76·5-s − 0.197·6-s + 0.626·7-s + 0.353·8-s − 0.922·9-s − 1.24·10-s + 1.03·11-s − 0.139·12-s − 0.951·13-s + 0.442·14-s + 0.491·15-s + 0.250·16-s + 0.873·17-s − 0.652·18-s − 0.770·19-s − 0.880·20-s − 0.174·21-s + 0.729·22-s + 1.60·23-s − 0.0986·24-s + 2.10·25-s − 0.672·26-s + 0.536·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + 1.44T + 27T^{2} \) |
| 5 | \( 1 + 19.6T + 125T^{2} \) |
| 7 | \( 1 - 11.5T + 343T^{2} \) |
| 11 | \( 1 - 37.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 233.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 10.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 14.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 606.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 702.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 543.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 314.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 859.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 725.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 820.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 648.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 60.9T + 9.12e5T^{2} \) |
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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
−8.357459420570986983430868565558, −7.76988079725919153404944185418, −6.97720122297345541946155170602, −6.19204900942041565699404116757, −4.89296138084937115813747441373, −4.63409825890299690505222187276, −3.51591828231021456502152134984, −2.85333699546830435676548938648, −1.21611232630329423102493202622, 0,
1.21611232630329423102493202622, 2.85333699546830435676548938648, 3.51591828231021456502152134984, 4.63409825890299690505222187276, 4.89296138084937115813747441373, 6.19204900942041565699404116757, 6.97720122297345541946155170602, 7.76988079725919153404944185418, 8.357459420570986983430868565558