L(s) = 1 | + 5-s − 3·9-s − 4·11-s − 6·13-s − 6·17-s − 19-s + 8·23-s + 25-s − 2·29-s + 2·37-s + 2·41-s + 4·43-s − 3·45-s − 8·47-s − 7·49-s − 6·53-s − 4·55-s − 4·59-s − 2·61-s − 6·65-s + 8·67-s + 8·71-s + 2·73-s − 8·79-s + 9·81-s + 4·83-s − 6·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.229·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.447·45-s − 1.16·47-s − 49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.744·65-s + 0.977·67-s + 0.949·71-s + 0.234·73-s − 0.900·79-s + 81-s + 0.439·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.826216841533732797390026366053, −9.135414516724356145082561336474, −8.224600959705339308697019103183, −7.30913391702283676603688628042, −6.39349606065152247841179289575, −5.25613153784368692505455427327, −4.73124222491170734087207238309, −2.95824527652601928276332710598, −2.27632682552704718607324575677, 0,
2.27632682552704718607324575677, 2.95824527652601928276332710598, 4.73124222491170734087207238309, 5.25613153784368692505455427327, 6.39349606065152247841179289575, 7.30913391702283676603688628042, 8.224600959705339308697019103183, 9.135414516724356145082561336474, 9.826216841533732797390026366053