L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·11-s + 4·13-s + 15-s − 6·17-s + 2·19-s − 21-s + 23-s + 25-s + 27-s − 6·29-s + 2·31-s − 2·33-s − 35-s − 10·37-s + 4·39-s − 6·41-s + 4·43-s + 45-s + 8·47-s + 49-s − 6·51-s − 2·55-s + 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s − 0.169·35-s − 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.269·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.46977998429466, −13.23222605316512, −12.79033312660222, −12.21835887673729, −11.61508760508492, −10.98394343070590, −10.79098479775555, −10.07842954113379, −9.815428623512668, −8.988834166268986, −8.797312529625848, −8.495369302104197, −7.624982701426969, −7.267138329308590, −6.725993847068032, −6.173496638425949, −5.715788307515787, −5.083268933413171, −4.573000557281198, −3.792864833451213, −3.488744938142402, −2.792479604983207, −2.181423177694044, −1.724260240299504, −0.8883751522166843, 0,
0.8883751522166843, 1.724260240299504, 2.181423177694044, 2.792479604983207, 3.488744938142402, 3.792864833451213, 4.573000557281198, 5.083268933413171, 5.715788307515787, 6.173496638425949, 6.725993847068032, 7.267138329308590, 7.624982701426969, 8.495369302104197, 8.797312529625848, 8.988834166268986, 9.815428623512668, 10.07842954113379, 10.79098479775555, 10.98394343070590, 11.61508760508492, 12.21835887673729, 12.79033312660222, 13.23222605316512, 13.46977998429466