L(s) = 1 | − 1.26·2-s + 0.450·3-s − 0.412·4-s + 1.25·5-s − 0.568·6-s + 3.91·7-s + 3.03·8-s − 2.79·9-s − 1.58·10-s − 11-s − 0.185·12-s − 4.06·13-s − 4.93·14-s + 0.567·15-s − 3.00·16-s − 4.44·17-s + 3.52·18-s − 2.43·19-s − 0.519·20-s + 1.76·21-s + 1.26·22-s + 0.395·23-s + 1.37·24-s − 3.41·25-s + 5.12·26-s − 2.61·27-s − 1.61·28-s + ⋯ |
L(s) = 1 | − 0.890·2-s + 0.260·3-s − 0.206·4-s + 0.563·5-s − 0.231·6-s + 1.47·7-s + 1.07·8-s − 0.932·9-s − 0.501·10-s − 0.301·11-s − 0.0536·12-s − 1.12·13-s − 1.31·14-s + 0.146·15-s − 0.751·16-s − 1.07·17-s + 0.830·18-s − 0.558·19-s − 0.116·20-s + 0.385·21-s + 0.268·22-s + 0.0825·23-s + 0.279·24-s − 0.682·25-s + 1.00·26-s − 0.503·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 - 0.450T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 13 | \( 1 + 4.06T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 - 0.395T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 - 9.76T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 - 5.60T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 - 0.203T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 + 9.76T + 97T^{2} \) |
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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
−9.010625749205666388769936864388, −8.447711956815855121607187911091, −7.80479470016669688818809626741, −7.03508123837579838585232783316, −5.65427992986139318293523554339, −4.98291217720287534102514819787, −4.14044992859514901062243744924, −2.43590809624747968167459460364, −1.75101713406692439030949905932, 0,
1.75101713406692439030949905932, 2.43590809624747968167459460364, 4.14044992859514901062243744924, 4.98291217720287534102514819787, 5.65427992986139318293523554339, 7.03508123837579838585232783316, 7.80479470016669688818809626741, 8.447711956815855121607187911091, 9.010625749205666388769936864388