| L(s) = 1 | − 3·3-s − 4·5-s − 7-s + 6·9-s − 6·11-s − 7·13-s + 12·15-s − 7·17-s − 8·19-s + 3·21-s − 6·23-s + 11·25-s − 9·27-s − 2·29-s − 7·31-s + 18·33-s + 4·35-s − 7·37-s + 21·39-s + 5·41-s − 24·45-s − 6·47-s + 49-s + 21·51-s − 10·53-s + 24·55-s + 24·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s − 0.377·7-s + 2·9-s − 1.80·11-s − 1.94·13-s + 3.09·15-s − 1.69·17-s − 1.83·19-s + 0.654·21-s − 1.25·23-s + 11/5·25-s − 1.73·27-s − 0.371·29-s − 1.25·31-s + 3.13·33-s + 0.676·35-s − 1.15·37-s + 3.36·39-s + 0.780·41-s − 3.57·45-s − 0.875·47-s + 1/7·49-s + 2.94·51-s − 1.37·53-s + 3.23·55-s + 3.17·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 373 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.19489529551934, −15.99793786494341, −15.55476631080825, −15.02858101398039, −14.56101387820836, −13.32691029240681, −12.79832100174773, −12.49387177454269, −12.14312752972051, −11.42108193656213, −10.90683508873543, −10.67629146886166, −10.09212007627868, −9.296416072218432, −8.358086390794446, −7.874134359105098, −7.234981728128187, −6.884069048247837, −6.197949378574714, −5.353256698872530, −4.811515052989155, −4.425673245136290, −3.817320072621513, −2.661579423864415, −1.991679235911286, 0, 0, 0,
1.991679235911286, 2.661579423864415, 3.817320072621513, 4.425673245136290, 4.811515052989155, 5.353256698872530, 6.197949378574714, 6.884069048247837, 7.234981728128187, 7.874134359105098, 8.358086390794446, 9.296416072218432, 10.09212007627868, 10.67629146886166, 10.90683508873543, 11.42108193656213, 12.14312752972051, 12.49387177454269, 12.79832100174773, 13.32691029240681, 14.56101387820836, 15.02858101398039, 15.55476631080825, 15.99793786494341, 16.19489529551934
