L(s) = 1 | − 0.700·3-s − 5-s + 7-s − 2.50·9-s + 4.66·11-s + 13-s + 0.700·15-s − 5.15·17-s − 1.75·19-s − 0.700·21-s + 4.06·23-s + 25-s + 3.85·27-s − 7.96·29-s − 1.99·31-s − 3.26·33-s − 35-s − 8.63·37-s − 0.700·39-s + 7.22·41-s − 7.71·43-s + 2.50·45-s + 3.96·47-s + 49-s + 3.61·51-s + 10.7·53-s − 4.66·55-s + ⋯ |
L(s) = 1 | − 0.404·3-s − 0.447·5-s + 0.377·7-s − 0.836·9-s + 1.40·11-s + 0.277·13-s + 0.180·15-s − 1.25·17-s − 0.403·19-s − 0.152·21-s + 0.848·23-s + 0.200·25-s + 0.742·27-s − 1.47·29-s − 0.358·31-s − 0.568·33-s − 0.169·35-s − 1.41·37-s − 0.112·39-s + 1.12·41-s − 1.17·43-s + 0.374·45-s + 0.577·47-s + 0.142·49-s + 0.505·51-s + 1.47·53-s − 0.629·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 0.700T + 3T^{2} \) |
| 11 | \( 1 - 4.66T + 11T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 + 1.99T + 31T^{2} \) |
| 37 | \( 1 + 8.63T + 37T^{2} \) |
| 41 | \( 1 - 7.22T + 41T^{2} \) |
| 43 | \( 1 + 7.71T + 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.22T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 2.24T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 + 5.40T + 83T^{2} \) |
| 89 | \( 1 + 8.98T + 89T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
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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
−8.448183344410665404530740630882, −7.19483944615982863004627126913, −6.80405467068409336760536802224, −5.89862804977074013086616778843, −5.21323389393103499538876541424, −4.22321341266488313414382565560, −3.66709176799971323918332489009, −2.47618769310528413117324280590, −1.35637210181047361186257302461, 0,
1.35637210181047361186257302461, 2.47618769310528413117324280590, 3.66709176799971323918332489009, 4.22321341266488313414382565560, 5.21323389393103499538876541424, 5.89862804977074013086616778843, 6.80405467068409336760536802224, 7.19483944615982863004627126913, 8.448183344410665404530740630882