L(s) = 1 | + 1.58·2-s − 5.48·4-s − 5·5-s − 21.3·8-s − 7.92·10-s − 28.1·11-s − 3.85·13-s + 9.97·16-s − 38.3·17-s + 116.·19-s + 27.4·20-s − 44.6·22-s + 176.·23-s + 25·25-s − 6.11·26-s + 209.·29-s − 207.·31-s + 186.·32-s − 60.8·34-s − 15.6·37-s + 185.·38-s + 106.·40-s + 10.5·41-s − 325.·43-s + 154.·44-s + 279.·46-s + 188.·47-s + ⋯ |
L(s) = 1 | + 0.560·2-s − 0.685·4-s − 0.447·5-s − 0.945·8-s − 0.250·10-s − 0.771·11-s − 0.0823·13-s + 0.155·16-s − 0.547·17-s + 1.40·19-s + 0.306·20-s − 0.432·22-s + 1.59·23-s + 0.200·25-s − 0.0461·26-s + 1.34·29-s − 1.20·31-s + 1.03·32-s − 0.306·34-s − 0.0696·37-s + 0.790·38-s + 0.422·40-s + 0.0402·41-s − 1.15·43-s + 0.528·44-s + 0.896·46-s + 0.585·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.58T + 8T^{2} \) |
| 11 | \( 1 + 28.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.85T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 15.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 43.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 855.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 252.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 960.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.436215232052288170605157929275, −7.47383356763882272381066987719, −6.81178738910464387806280609227, −5.60966045456367594774020552938, −5.09411891075735387140374053414, −4.35567888108889052453068220515, −3.35878262868964406708354641332, −2.72213243443484472799642330700, −1.07448971561812956983625369040, 0,
1.07448971561812956983625369040, 2.72213243443484472799642330700, 3.35878262868964406708354641332, 4.35567888108889052453068220515, 5.09411891075735387140374053414, 5.60966045456367594774020552938, 6.81178738910464387806280609227, 7.47383356763882272381066987719, 8.436215232052288170605157929275