L(s) = 1 | − 2-s − 4-s + 3·5-s − 7-s + 3·8-s − 3·10-s − 11-s + 2·13-s + 14-s − 16-s − 4·17-s − 3·20-s + 22-s + 3·23-s + 4·25-s − 2·26-s + 28-s − 8·29-s − 2·31-s − 5·32-s + 4·34-s − 3·35-s + 2·37-s + 9·40-s + 9·41-s + 11·43-s + 44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.34·5-s − 0.377·7-s + 1.06·8-s − 0.948·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.670·20-s + 0.213·22-s + 0.625·23-s + 4/5·25-s − 0.392·26-s + 0.188·28-s − 1.48·29-s − 0.359·31-s − 0.883·32-s + 0.685·34-s − 0.507·35-s + 0.328·37-s + 1.42·40-s + 1.40·41-s + 1.67·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.06247865308167, −12.83631155009023, −12.39310387268508, −11.44983329102912, −10.98505950974328, −10.72055678488926, −10.25022392158402, −9.713311616231831, −9.241593003876055, −8.973961400198871, −8.826163652624938, −7.859702280657012, −7.511507016460823, −7.107006269678360, −6.292282387622571, −5.935374095945476, −5.602414651819871, −4.922553544672705, −4.379673600689298, −3.920028151224568, −3.185376249270979, −2.454186151091864, −2.056838126215557, −1.379405336647101, −0.7925452685285312, 0,
0.7925452685285312, 1.379405336647101, 2.056838126215557, 2.454186151091864, 3.185376249270979, 3.920028151224568, 4.379673600689298, 4.922553544672705, 5.602414651819871, 5.935374095945476, 6.292282387622571, 7.107006269678360, 7.511507016460823, 7.859702280657012, 8.826163652624938, 8.973961400198871, 9.241593003876055, 9.713311616231831, 10.25022392158402, 10.72055678488926, 10.98505950974328, 11.44983329102912, 12.39310387268508, 12.83631155009023, 13.06247865308167