L(s) = 1 | + 0.121·3-s + 5-s + 5.25·7-s − 2.98·9-s − 2.85·11-s − 2.22·13-s + 0.121·15-s − 7.47·17-s − 0.221·19-s + 0.640·21-s + 1.75·23-s + 25-s − 0.728·27-s + 0.610·29-s − 3.10·31-s − 0.347·33-s + 5.25·35-s + 1.19·37-s − 0.270·39-s + 4.15·41-s + 7.47·43-s − 2.98·45-s − 10.1·47-s + 20.6·49-s − 0.909·51-s + 6.80·53-s − 2.85·55-s + ⋯ |
L(s) = 1 | + 0.0702·3-s + 0.447·5-s + 1.98·7-s − 0.995·9-s − 0.859·11-s − 0.615·13-s + 0.0314·15-s − 1.81·17-s − 0.0507·19-s + 0.139·21-s + 0.366·23-s + 0.200·25-s − 0.140·27-s + 0.113·29-s − 0.557·31-s − 0.0604·33-s + 0.888·35-s + 0.197·37-s − 0.0432·39-s + 0.649·41-s + 1.13·43-s − 0.445·45-s − 1.47·47-s + 2.94·49-s − 0.127·51-s + 0.935·53-s − 0.384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 0.121T + 3T^{2} \) |
| 7 | \( 1 - 5.25T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 + 0.221T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 0.610T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 - 7.47T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 - 0.527T + 59T^{2} \) |
| 61 | \( 1 + 0.958T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 + 8.66T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 - 0.669T + 97T^{2} \) |
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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
−7.55425912659999061046064689578, −6.94779050782514479492437978224, −5.87641439485979180046751874370, −5.36617476566279254077617579166, −4.72485661383385003577722650748, −4.18436468479210363483717348940, −2.70056254735163595859009415720, −2.34726287356850608979536852887, −1.43591191260486724022384542832, 0,
1.43591191260486724022384542832, 2.34726287356850608979536852887, 2.70056254735163595859009415720, 4.18436468479210363483717348940, 4.72485661383385003577722650748, 5.36617476566279254077617579166, 5.87641439485979180046751874370, 6.94779050782514479492437978224, 7.55425912659999061046064689578