L(s) = 1 | + 2-s + 1.71·3-s + 4-s + 2.08·5-s + 1.71·6-s − 3.84·7-s + 8-s − 0.0606·9-s + 2.08·10-s − 4.60·11-s + 1.71·12-s + 1.14·13-s − 3.84·14-s + 3.57·15-s + 16-s − 3.79·17-s − 0.0606·18-s − 2.00·19-s + 2.08·20-s − 6.59·21-s − 4.60·22-s + 1.16·23-s + 1.71·24-s − 0.646·25-s + 1.14·26-s − 5.24·27-s − 3.84·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.989·3-s + 0.5·4-s + 0.933·5-s + 0.699·6-s − 1.45·7-s + 0.353·8-s − 0.0202·9-s + 0.659·10-s − 1.38·11-s + 0.494·12-s + 0.316·13-s − 1.02·14-s + 0.923·15-s + 0.250·16-s − 0.921·17-s − 0.0142·18-s − 0.460·19-s + 0.466·20-s − 1.43·21-s − 0.981·22-s + 0.242·23-s + 0.349·24-s − 0.129·25-s + 0.223·26-s − 1.00·27-s − 0.727·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 2.00T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 0.335T + 29T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 + 1.50T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 7.71T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 2.36T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
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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
−7.68847730144575393872860431092, −6.89546680022811776059774411534, −6.14436173071530217473928112954, −5.75340077255570691343911511389, −4.80921098736762973442106821618, −3.87407480357628910394491190144, −2.98429445872514786409746572710, −2.66379388721638301686730682362, −1.85185679930903283603916612331, 0,
1.85185679930903283603916612331, 2.66379388721638301686730682362, 2.98429445872514786409746572710, 3.87407480357628910394491190144, 4.80921098736762973442106821618, 5.75340077255570691343911511389, 6.14436173071530217473928112954, 6.89546680022811776059774411534, 7.68847730144575393872860431092