L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 11-s + 4·13-s + 15-s − 6·17-s − 7·19-s + 21-s + 25-s − 27-s − 6·29-s + 4·31-s − 33-s + 35-s + 2·37-s − 4·39-s + 9·41-s − 2·43-s − 45-s + 7·47-s + 49-s + 6·51-s − 5·53-s − 55-s + 7·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s − 1.60·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s + 0.169·35-s + 0.328·37-s − 0.640·39-s + 1.40·41-s − 0.304·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s + 0.840·51-s − 0.686·53-s − 0.134·55-s + 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.17927137166538, −12.64057080414671, −12.45146668316865, −11.58443088044780, −11.32638558063551, −10.93892996896942, −10.62166544934950, −9.949943413262520, −9.465358669181464, −8.840790625117116, −8.565869580930900, −8.130159153165242, −7.287187071391950, −7.015531729383833, −6.394285127978084, −6.053236480964000, −5.698646789614962, −4.763569900245837, −4.406739521899256, −3.915925246267528, −3.554585221400674, −2.582953717180320, −2.186524943288300, −1.395966451927177, −0.6615844048850001, 0,
0.6615844048850001, 1.395966451927177, 2.186524943288300, 2.582953717180320, 3.554585221400674, 3.915925246267528, 4.406739521899256, 4.763569900245837, 5.698646789614962, 6.053236480964000, 6.394285127978084, 7.015531729383833, 7.287187071391950, 8.130159153165242, 8.565869580930900, 8.840790625117116, 9.465358669181464, 9.949943413262520, 10.62166544934950, 10.93892996896942, 11.32638558063551, 11.58443088044780, 12.45146668316865, 12.64057080414671, 13.17927137166538