L(s) = 1 | − 5-s + 7-s − 6·11-s + 6·13-s − 2·17-s + 7·19-s − 23-s − 4·25-s + 2·29-s + 10·31-s − 35-s − 6·37-s − 8·41-s − 10·43-s + 8·47-s + 49-s + 2·53-s + 6·55-s + 7·61-s − 6·65-s − 12·67-s + 15·71-s − 2·73-s − 6·77-s + 79-s + 12·83-s + 2·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.80·11-s + 1.66·13-s − 0.485·17-s + 1.60·19-s − 0.208·23-s − 4/5·25-s + 0.371·29-s + 1.79·31-s − 0.169·35-s − 0.986·37-s − 1.24·41-s − 1.52·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s + 0.809·55-s + 0.896·61-s − 0.744·65-s − 1.46·67-s + 1.78·71-s − 0.234·73-s − 0.683·77-s + 0.112·79-s + 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.661350015\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.661350015\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.258534877886123881516881958726, −7.79067111123171966699020607366, −6.96995072676846735892694754372, −6.08177989229326585475743736013, −5.31878813596332244967742727497, −4.72715111248749756023836609092, −3.66159139267585105415203405545, −3.02811103455129536867830610709, −1.92802436486972449904496470791, −0.72081180776643933088668872598,
0.72081180776643933088668872598, 1.92802436486972449904496470791, 3.02811103455129536867830610709, 3.66159139267585105415203405545, 4.72715111248749756023836609092, 5.31878813596332244967742727497, 6.08177989229326585475743736013, 6.96995072676846735892694754372, 7.79067111123171966699020607366, 8.258534877886123881516881958726