| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 9-s − 4·13-s + 4·15-s + 2·17-s − 2·21-s + 23-s − 25-s − 4·27-s + 2·29-s − 2·35-s − 4·37-s − 8·39-s − 6·41-s − 2·43-s + 2·45-s + 4·47-s + 49-s + 4·51-s + 2·59-s + 10·61-s − 63-s − 8·65-s + 2·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 1.03·15-s + 0.485·17-s − 0.436·21-s + 0.208·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.338·35-s − 0.657·37-s − 1.28·39-s − 0.937·41-s − 0.304·43-s + 0.298·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.260·59-s + 1.28·61-s − 0.125·63-s − 0.992·65-s + 0.244·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.63590188422244, −13.10010125251045, −12.83115846390555, −12.13281406589875, −11.77026789390819, −11.19216365004094, −10.31946267433399, −10.06703023787049, −9.802620176109602, −9.118041994008854, −8.847912150875427, −8.347764286743743, −7.604215581399192, −7.439789307625583, −6.690809665167619, −6.241017489710490, −5.576916034835130, −5.146222510325980, −4.587961479818766, −3.719394851839909, −3.389182233529330, −2.735658659320169, −2.222429627158237, −1.869110923259582, −0.9528903618977068, 0,
0.9528903618977068, 1.869110923259582, 2.222429627158237, 2.735658659320169, 3.389182233529330, 3.719394851839909, 4.587961479818766, 5.146222510325980, 5.576916034835130, 6.241017489710490, 6.690809665167619, 7.439789307625583, 7.604215581399192, 8.347764286743743, 8.847912150875427, 9.118041994008854, 9.802620176109602, 10.06703023787049, 10.31946267433399, 11.19216365004094, 11.77026789390819, 12.13281406589875, 12.83115846390555, 13.10010125251045, 13.63590188422244
