L(s) = 1 | − 7-s + 5·11-s − 13-s − 3·17-s + 4·23-s + 9·29-s − 7·31-s + 8·37-s + 2·41-s − 7·47-s − 6·49-s − 3·53-s − 9·59-s + 15·61-s + 7·67-s − 4·73-s − 5·77-s − 8·79-s + 7·83-s + 12·89-s + 91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.50·11-s − 0.277·13-s − 0.727·17-s + 0.834·23-s + 1.67·29-s − 1.25·31-s + 1.31·37-s + 0.312·41-s − 1.02·47-s − 6/7·49-s − 0.412·53-s − 1.17·59-s + 1.92·61-s + 0.855·67-s − 0.468·73-s − 0.569·77-s − 0.900·79-s + 0.768·83-s + 1.27·89-s + 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.442718022\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.442718022\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.00783238605268, −12.72176765647274, −12.21568579764640, −11.67277205477118, −11.19187167313721, −11.00441919806237, −10.17491671149447, −9.762832594880104, −9.321550139561865, −8.930365362194513, −8.441066336749147, −7.882529696213651, −7.233426325211327, −6.760781906343315, −6.390260346052540, −6.038490739164474, −5.119975528806237, −4.819786900984646, −4.094254401562348, −3.769702413617448, −3.007533618832024, −2.579024641656187, −1.748433577173964, −1.193937966976724, −0.4735361525703584,
0.4735361525703584, 1.193937966976724, 1.748433577173964, 2.579024641656187, 3.007533618832024, 3.769702413617448, 4.094254401562348, 4.819786900984646, 5.119975528806237, 6.038490739164474, 6.390260346052540, 6.760781906343315, 7.233426325211327, 7.882529696213651, 8.441066336749147, 8.930365362194513, 9.321550139561865, 9.762832594880104, 10.17491671149447, 11.00441919806237, 11.19187167313721, 11.67277205477118, 12.21568579764640, 12.72176765647274, 13.00783238605268