L(s) = 1 | − 2-s + 2.50·3-s + 4-s + 0.0842·5-s − 2.50·6-s − 0.974·7-s − 8-s + 3.27·9-s − 0.0842·10-s − 2.63·11-s + 2.50·12-s + 2.36·13-s + 0.974·14-s + 0.211·15-s + 16-s − 4.30·17-s − 3.27·18-s + 5.00·19-s + 0.0842·20-s − 2.44·21-s + 2.63·22-s − 4.69·23-s − 2.50·24-s − 4.99·25-s − 2.36·26-s + 0.694·27-s − 0.974·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.44·3-s + 0.5·4-s + 0.0376·5-s − 1.02·6-s − 0.368·7-s − 0.353·8-s + 1.09·9-s − 0.0266·10-s − 0.795·11-s + 0.723·12-s + 0.655·13-s + 0.260·14-s + 0.0545·15-s + 0.250·16-s − 1.04·17-s − 0.772·18-s + 1.14·19-s + 0.0188·20-s − 0.532·21-s + 0.562·22-s − 0.979·23-s − 0.511·24-s − 0.998·25-s − 0.463·26-s + 0.133·27-s − 0.184·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 - 0.0842T + 5T^{2} \) |
| 7 | \( 1 + 0.974T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 - 0.573T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 7.80T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 - 4.39T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 - 8.84T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 3.01T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 - 9.95T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−8.426622387529036789124756725128, −7.44183135751537867141187232106, −7.04107512673920132957816774568, −5.97194969211745308835150878148, −5.13676600035122073553219337666, −3.75276772171368745659789646147, −3.38879138703116869865787751834, −2.31262016865217238248823933817, −1.72747296504051743189023366707, 0,
1.72747296504051743189023366707, 2.31262016865217238248823933817, 3.38879138703116869865787751834, 3.75276772171368745659789646147, 5.13676600035122073553219337666, 5.97194969211745308835150878148, 7.04107512673920132957816774568, 7.44183135751537867141187232106, 8.426622387529036789124756725128