| L(s) = 1 | − 5-s + 11-s + 5·13-s + 17-s + 6·19-s + 4·23-s + 25-s − 3·29-s − 2·31-s + 8·37-s − 10·41-s − 2·43-s − 7·47-s + 2·53-s − 55-s + 14·59-s + 8·61-s − 5·65-s + 14·67-s + 10·73-s − 11·79-s − 4·83-s − 85-s + 4·89-s − 6·95-s + 3·97-s − 10·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s + 1.38·13-s + 0.242·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.557·29-s − 0.359·31-s + 1.31·37-s − 1.56·41-s − 0.304·43-s − 1.02·47-s + 0.274·53-s − 0.134·55-s + 1.82·59-s + 1.02·61-s − 0.620·65-s + 1.71·67-s + 1.17·73-s − 1.23·79-s − 0.439·83-s − 0.108·85-s + 0.423·89-s − 0.615·95-s + 0.304·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.256456748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256456748\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−7.897896371463336345266991990935, −6.92744854755746429975089901874, −6.59474597232800625069678245466, −5.51589090779766153294605834687, −5.17174879937803300953760703666, −4.00544317776361456006542139262, −3.58722367743226563989338538796, −2.79857499378462339469274671475, −1.55783645740042646619385285491, −0.78870727948012110458706875692,
0.78870727948012110458706875692, 1.55783645740042646619385285491, 2.79857499378462339469274671475, 3.58722367743226563989338538796, 4.00544317776361456006542139262, 5.17174879937803300953760703666, 5.51589090779766153294605834687, 6.59474597232800625069678245466, 6.92744854755746429975089901874, 7.897896371463336345266991990935
