| L(s) = 1 | − 1.41·2-s + 5-s + 2.82·8-s − 1.41·10-s + 5.82·11-s − 1.58·13-s − 4.00·16-s − 5.24·17-s − 6·19-s − 8.24·22-s − 4.58·23-s + 25-s + 2.24·26-s − 2.65·29-s − 1.75·31-s + 7.41·34-s − 6.24·37-s + 8.48·38-s + 2.82·40-s + 2.24·41-s + 2·43-s + 6.48·46-s + 1.24·47-s − 1.41·50-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.447·5-s + 0.999·8-s − 0.447·10-s + 1.75·11-s − 0.439·13-s − 1.00·16-s − 1.27·17-s − 1.37·19-s − 1.75·22-s − 0.956·23-s + 0.200·25-s + 0.439·26-s − 0.493·29-s − 0.315·31-s + 1.27·34-s − 1.02·37-s + 1.37·38-s + 0.447·40-s + 0.350·41-s + 0.304·43-s + 0.956·46-s + 0.181·47-s − 0.200·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 1.24T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 0.242T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−8.838014380052564093404077658087, −8.200162974778537628461116294736, −7.05068992745308560988020065391, −6.62910725745994479927627913237, −5.63500727639793065082853805819, −4.38628701235193445689180419820, −3.97834770370884018687487707918, −2.26331358665971321584744309513, −1.49021843420171989920473165638, 0,
1.49021843420171989920473165638, 2.26331358665971321584744309513, 3.97834770370884018687487707918, 4.38628701235193445689180419820, 5.63500727639793065082853805819, 6.62910725745994479927627913237, 7.05068992745308560988020065391, 8.200162974778537628461116294736, 8.838014380052564093404077658087
