L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 11-s − 6·13-s − 2·14-s + 16-s + 17-s + 6·19-s + 22-s + 5·23-s − 6·26-s − 2·28-s + 9·29-s − 4·31-s + 32-s + 34-s − 8·37-s + 6·38-s − 8·43-s + 44-s + 5·46-s − 3·49-s − 6·52-s − 4·53-s − 2·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.301·11-s − 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 1.37·19-s + 0.213·22-s + 1.04·23-s − 1.17·26-s − 0.377·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 1.31·37-s + 0.973·38-s − 1.21·43-s + 0.150·44-s + 0.737·46-s − 3/7·49-s − 0.832·52-s − 0.549·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.36764436353839, −14.25570911781019, −13.66118135134447, −13.04801410250572, −12.55199885977721, −12.19847948493118, −11.74918660041862, −11.21072542898528, −10.48266649480383, −9.965990902800923, −9.624723705256638, −9.064006495768281, −8.330953377925172, −7.668231814713632, −7.083605291672855, −6.805739371122522, −6.200921005283252, −5.338780091346938, −5.049207347101570, −4.557947004801955, −3.593315100065533, −3.162530171702216, −2.712378875864453, −1.849264022413235, −1.010868942412319, 0,
1.010868942412319, 1.849264022413235, 2.712378875864453, 3.162530171702216, 3.593315100065533, 4.557947004801955, 5.049207347101570, 5.338780091346938, 6.200921005283252, 6.805739371122522, 7.083605291672855, 7.668231814713632, 8.330953377925172, 9.064006495768281, 9.624723705256638, 9.965990902800923, 10.48266649480383, 11.21072542898528, 11.74918660041862, 12.19847948493118, 12.55199885977721, 13.04801410250572, 13.66118135134447, 14.25570911781019, 14.36764436353839