| L(s) = 1 | + 2-s − 4-s − 2·5-s − 7-s − 3·8-s − 2·10-s + 2·13-s − 14-s − 16-s + 4·17-s − 2·19-s + 2·20-s − 25-s + 2·26-s + 28-s − 29-s − 2·31-s + 5·32-s + 4·34-s + 2·35-s + 2·37-s − 2·38-s + 6·40-s + 10·47-s + 49-s − 50-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s − 1.06·8-s − 0.632·10-s + 0.554·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.447·20-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.185·29-s − 0.359·31-s + 0.883·32-s + 0.685·34-s + 0.338·35-s + 0.328·37-s − 0.324·38-s + 0.948·40-s + 1.45·47-s + 1/7·49-s − 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.01923873029766, −12.75297011315431, −12.43346386355525, −11.91809798127909, −11.40322615228391, −11.08954219792720, −10.38916811865460, −9.919433040363163, −9.446771356874640, −8.925976338533249, −8.492043731475766, −7.971920506266217, −7.580786522887133, −6.970904191395546, −6.365823929492640, −5.859297097995105, −5.511738721440354, −4.857986250672533, −4.305081647528642, −3.871838225078775, −3.504456524945811, −2.986409789861602, −2.331493115275697, −1.399661331499403, −0.6700716939160224, 0,
0.6700716939160224, 1.399661331499403, 2.331493115275697, 2.986409789861602, 3.504456524945811, 3.871838225078775, 4.305081647528642, 4.857986250672533, 5.511738721440354, 5.859297097995105, 6.365823929492640, 6.970904191395546, 7.580786522887133, 7.971920506266217, 8.492043731475766, 8.925976338533249, 9.446771356874640, 9.919433040363163, 10.38916811865460, 11.08954219792720, 11.40322615228391, 11.91809798127909, 12.43346386355525, 12.75297011315431, 13.01923873029766
