| L(s) = 1 | − 2-s − 2·3-s + 4-s − 5-s + 2·6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 2·12-s + 13-s − 14-s + 2·15-s + 16-s − 18-s + 2·19-s − 20-s − 2·21-s − 22-s − 6·23-s + 2·24-s + 25-s − 26-s + 4·27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.577·12-s + 0.277·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s + 0.769·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6077165723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6077165723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−16.70884151650296, −16.26060396360027, −15.72321303515505, −15.11866385038805, −14.38495368340942, −13.89628110373657, −12.92536736626161, −12.31573225186273, −11.85243150159084, −11.27802248797989, −10.93672301937364, −10.33961909242618, −9.531051816438761, −9.019346669262517, −8.182026892917682, −7.679627749503156, −7.026148798399761, −6.251533512973244, −5.753208979239760, −5.097913658790192, −4.195841229052003, −3.506310872153976, −2.359612470572262, −1.404268135002306, −0.4651502728336448,
0.4651502728336448, 1.404268135002306, 2.359612470572262, 3.506310872153976, 4.195841229052003, 5.097913658790192, 5.753208979239760, 6.251533512973244, 7.026148798399761, 7.679627749503156, 8.182026892917682, 9.019346669262517, 9.531051816438761, 10.33961909242618, 10.93672301937364, 11.27802248797989, 11.85243150159084, 12.31573225186273, 12.92536736626161, 13.89628110373657, 14.38495368340942, 15.11866385038805, 15.72321303515505, 16.26060396360027, 16.70884151650296
