L(s) = 1 | − 1.41·2-s + 3-s − 2.41·5-s − 1.41·6-s + 7-s + 2.82·8-s + 9-s + 3.41·10-s + 5.65·11-s − 1.41·14-s − 2.41·15-s − 4.00·16-s + 17-s − 1.41·18-s − 7.82·19-s + 21-s − 8.00·22-s − 7.24·23-s + 2.82·24-s + 0.828·25-s + 27-s − 0.828·29-s + 3.41·30-s + 9.65·31-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 0.577·3-s − 1.07·5-s − 0.577·6-s + 0.377·7-s + 0.999·8-s + 0.333·9-s + 1.07·10-s + 1.70·11-s − 0.377·14-s − 0.623·15-s − 1.00·16-s + 0.242·17-s − 0.333·18-s − 1.79·19-s + 0.218·21-s − 1.70·22-s − 1.51·23-s + 0.577·24-s + 0.165·25-s + 0.192·27-s − 0.153·29-s + 0.623·30-s + 1.73·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 7.82T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 9.65T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 4.07T + 47T^{2} \) |
| 53 | \( 1 + 9.58T + 53T^{2} \) |
| 59 | \( 1 + 6.41T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 83 | \( 1 - 2.58T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.265395834249552639322132181131, −7.70292240234634113292200529227, −6.85193130662849459276157786156, −6.19323830807920932281814692863, −4.69498284059816075472090026346, −4.12908298892777774549381727150, −3.64957590584275959763457369839, −2.14834043955688921636054283977, −1.28838607629585746848245914722, 0,
1.28838607629585746848245914722, 2.14834043955688921636054283977, 3.64957590584275959763457369839, 4.12908298892777774549381727150, 4.69498284059816075472090026346, 6.19323830807920932281814692863, 6.85193130662849459276157786156, 7.70292240234634113292200529227, 8.265395834249552639322132181131