L(s) = 1 | − 2-s + 4-s − 2.82·5-s − 8-s + 2.82·10-s + 2·11-s + 16-s + 1.41·17-s − 7.07·19-s − 2.82·20-s − 2·22-s + 4·23-s + 3.00·25-s − 2·29-s + 8.48·31-s − 32-s − 1.41·34-s + 10·37-s + 7.07·38-s + 2.82·40-s + 9.89·41-s + 2·43-s + 2·44-s − 4·46-s − 2.82·47-s − 3.00·50-s + 2·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.26·5-s − 0.353·8-s + 0.894·10-s + 0.603·11-s + 0.250·16-s + 0.342·17-s − 1.62·19-s − 0.632·20-s − 0.426·22-s + 0.834·23-s + 0.600·25-s − 0.371·29-s + 1.52·31-s − 0.176·32-s − 0.242·34-s + 1.64·37-s + 1.14·38-s + 0.447·40-s + 1.54·41-s + 0.304·43-s + 0.301·44-s − 0.589·46-s − 0.412·47-s − 0.424·50-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8372555694\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8372555694\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−10.09862672275899663901852392365, −9.197385307516027437701847108737, −8.365171298233677812539811164957, −7.79859146866711010455637679558, −6.88060638619669399602089114130, −6.07270303300013490781867132820, −4.57508433531885028891043293554, −3.78906696726459253802734177551, −2.51699554533182135082823666397, −0.814577255811514319364394945426,
0.814577255811514319364394945426, 2.51699554533182135082823666397, 3.78906696726459253802734177551, 4.57508433531885028891043293554, 6.07270303300013490781867132820, 6.88060638619669399602089114130, 7.79859146866711010455637679558, 8.365171298233677812539811164957, 9.197385307516027437701847108737, 10.09862672275899663901852392365