L(s) = 1 | + 4·11-s + 2·13-s + 2·17-s + 19-s + 4·23-s + 6·29-s + 4·31-s − 6·37-s − 10·41-s − 4·43-s − 12·47-s − 7·49-s − 6·53-s − 12·59-s + 2·61-s + 4·67-s − 8·71-s + 6·73-s − 4·79-s + 12·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s + 1.11·29-s + 0.718·31-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s − 49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.450·79-s + 1.31·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.12411026087384, −12.34862226737305, −12.13962100860875, −11.58710489443088, −11.30616219889140, −10.68466727121458, −10.23298164098836, −9.710470104651949, −9.393074319905420, −8.739712950924802, −8.392840205427743, −8.014627296484275, −7.313859623469781, −6.692652255733004, −6.524136343498661, −6.054440867992387, −5.256959005322891, −4.852865914643999, −4.449439808933340, −3.616496137839539, −3.289884005766156, −2.906236589393143, −1.830183796887723, −1.487263003795853, −0.9130594715952859, 0,
0.9130594715952859, 1.487263003795853, 1.830183796887723, 2.906236589393143, 3.289884005766156, 3.616496137839539, 4.449439808933340, 4.852865914643999, 5.256959005322891, 6.054440867992387, 6.524136343498661, 6.692652255733004, 7.313859623469781, 8.014627296484275, 8.392840205427743, 8.739712950924802, 9.393074319905420, 9.710470104651949, 10.23298164098836, 10.68466727121458, 11.30616219889140, 11.58710489443088, 12.13962100860875, 12.34862226737305, 13.12411026087384