L(s) = 1 | + 5-s + 11-s − 6·13-s + 5·17-s + 2·19-s − 23-s − 4·25-s − 4·29-s + 2·31-s − 41-s + 8·43-s + 47-s − 10·53-s + 55-s + 6·59-s − 7·61-s − 6·65-s + 3·67-s + 2·73-s + 79-s + 9·83-s + 5·85-s − 16·89-s + 2·95-s + 11·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.301·11-s − 1.66·13-s + 1.21·17-s + 0.458·19-s − 0.208·23-s − 4/5·25-s − 0.742·29-s + 0.359·31-s − 0.156·41-s + 1.21·43-s + 0.145·47-s − 1.37·53-s + 0.134·55-s + 0.781·59-s − 0.896·61-s − 0.744·65-s + 0.366·67-s + 0.234·73-s + 0.112·79-s + 0.987·83-s + 0.542·85-s − 1.69·89-s + 0.205·95-s + 1.11·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.90284481004505, −14.58927473766573, −14.00017837978869, −13.73788490670392, −12.87626611663231, −12.44142285968546, −12.04181137968700, −11.50529042889137, −10.87579691684004, −10.15136382069166, −9.723832430971203, −9.496357382456797, −8.781619282395306, −7.910896281403077, −7.603489424490833, −7.100437134086851, −6.297693931507295, −5.757866995849283, −5.223240066253230, −4.656207749518448, −3.894032309886524, −3.216039397481858, −2.499872904738365, −1.880189609234399, −1.027569479464501, 0,
1.027569479464501, 1.880189609234399, 2.499872904738365, 3.216039397481858, 3.894032309886524, 4.656207749518448, 5.223240066253230, 5.757866995849283, 6.297693931507295, 7.100437134086851, 7.603489424490833, 7.910896281403077, 8.781619282395306, 9.496357382456797, 9.723832430971203, 10.15136382069166, 10.87579691684004, 11.50529042889137, 12.04181137968700, 12.44142285968546, 12.87626611663231, 13.73788490670392, 14.00017837978869, 14.58927473766573, 14.90284481004505