L(s) = 1 | + 0.670·2-s − 1.54·4-s + 2.54·5-s − 2.38·8-s + 1.71·10-s + 3.05·11-s − 13-s + 1.50·16-s + 1.34·17-s − 7.60·19-s − 3.95·20-s + 2.04·22-s − 1.84·23-s + 1.50·25-s − 0.670·26-s + 5.60·29-s − 10.2·31-s + 5.77·32-s + 0.900·34-s − 9.20·37-s − 5.09·38-s − 6.07·40-s − 8.04·41-s + 1.49·43-s − 4.73·44-s − 1.23·46-s − 3.89·47-s + ⋯ |
L(s) = 1 | + 0.474·2-s − 0.774·4-s + 1.14·5-s − 0.841·8-s + 0.540·10-s + 0.920·11-s − 0.277·13-s + 0.375·16-s + 0.325·17-s − 1.74·19-s − 0.883·20-s + 0.436·22-s − 0.384·23-s + 0.300·25-s − 0.131·26-s + 1.04·29-s − 1.84·31-s + 1.02·32-s + 0.154·34-s − 1.51·37-s − 0.827·38-s − 0.960·40-s − 1.25·41-s + 0.228·43-s − 0.713·44-s − 0.182·46-s − 0.567·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.670T + 2T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 17 | \( 1 - 1.34T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 + 1.84T + 23T^{2} \) |
| 29 | \( 1 - 5.60T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 9.20T + 37T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 - 1.49T + 43T^{2} \) |
| 47 | \( 1 + 3.89T + 47T^{2} \) |
| 53 | \( 1 + 0.502T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 - 0.683T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.91T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 7.18T + 97T^{2} \) |
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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
−7.87735713388101808109938553190, −6.62088663607660779326135894315, −6.40380694864835354492132734570, −5.45917633279771461146456915261, −4.99353186951057381344112732701, −4.04982139901124565441565025778, −3.47090250326482962683171520744, −2.28927227031623131388669179107, −1.50751898291570420595350312586, 0,
1.50751898291570420595350312586, 2.28927227031623131388669179107, 3.47090250326482962683171520744, 4.04982139901124565441565025778, 4.99353186951057381344112732701, 5.45917633279771461146456915261, 6.40380694864835354492132734570, 6.62088663607660779326135894315, 7.87735713388101808109938553190