L(s) = 1 | − 3-s − 5-s − 3.74·7-s + 9-s + 0.896·11-s − 1.29·13-s + 15-s − 4.92·17-s + 3.04·19-s + 3.74·21-s − 1.90·23-s + 25-s − 27-s + 8.22·29-s − 31-s − 0.896·33-s + 3.74·35-s + 8.78·37-s + 1.29·39-s + 10.0·41-s − 5.02·43-s − 45-s + 1.00·47-s + 7.03·49-s + 4.92·51-s − 2.01·53-s − 0.896·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.41·7-s + 0.333·9-s + 0.270·11-s − 0.358·13-s + 0.258·15-s − 1.19·17-s + 0.697·19-s + 0.817·21-s − 0.397·23-s + 0.200·25-s − 0.192·27-s + 1.52·29-s − 0.179·31-s − 0.155·33-s + 0.633·35-s + 1.44·37-s + 0.207·39-s + 1.57·41-s − 0.766·43-s − 0.149·45-s + 0.147·47-s + 1.00·49-s + 0.690·51-s − 0.276·53-s − 0.120·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 0.896T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 - 3.04T + 19T^{2} \) |
| 23 | \( 1 + 1.90T + 23T^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 5.02T + 43T^{2} \) |
| 47 | \( 1 - 1.00T + 47T^{2} \) |
| 53 | \( 1 + 2.01T + 53T^{2} \) |
| 59 | \( 1 + 2.28T + 59T^{2} \) |
| 61 | \( 1 - 9.93T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 - 3.73T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−7.41139859538187725515996729587, −6.67733953450792514606834862607, −6.33986000514217570120868070985, −5.55200288954676262910228253462, −4.58459501513714498465310298041, −4.06114194816075639556478356793, −3.12057949248167692643765937104, −2.41994508191683271554164130638, −0.973184252727378876839935444537, 0,
0.973184252727378876839935444537, 2.41994508191683271554164130638, 3.12057949248167692643765937104, 4.06114194816075639556478356793, 4.58459501513714498465310298041, 5.55200288954676262910228253462, 6.33986000514217570120868070985, 6.67733953450792514606834862607, 7.41139859538187725515996729587