L(s) = 1 | − 0.476·2-s + 3·3-s − 7.77·4-s + 17.0·5-s − 1.43·6-s − 27.8·7-s + 7.52·8-s + 9·9-s − 8.12·10-s − 11·11-s − 23.3·12-s + 32.3·13-s + 13.2·14-s + 51.1·15-s + 58.5·16-s − 67.3·17-s − 4.29·18-s + 107.·19-s − 132.·20-s − 83.4·21-s + 5.24·22-s − 79.8·23-s + 22.5·24-s + 165.·25-s − 15.4·26-s + 27·27-s + 216.·28-s + ⋯ |
L(s) = 1 | − 0.168·2-s + 0.577·3-s − 0.971·4-s + 1.52·5-s − 0.0973·6-s − 1.50·7-s + 0.332·8-s + 0.333·9-s − 0.256·10-s − 0.301·11-s − 0.560·12-s + 0.689·13-s + 0.253·14-s + 0.879·15-s + 0.915·16-s − 0.960·17-s − 0.0562·18-s + 1.29·19-s − 1.48·20-s − 0.866·21-s + 0.0508·22-s − 0.723·23-s + 0.191·24-s + 1.32·25-s − 0.116·26-s + 0.192·27-s + 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 + 0.476T + 8T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 7 | \( 1 + 27.8T + 343T^{2} \) |
| 13 | \( 1 - 32.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 67.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 328.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 73.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 382.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 144.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 618.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 577.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 265.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 805.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 940.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.72e3T + 9.12e5T^{2} \) |
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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
−8.799088134994071790982740711543, −7.70213284734941395996731595369, −6.78110350529806184072952302084, −5.89515373704576002244865266302, −5.42214484723679331344623848665, −4.14803271888916308179694144893, −3.33572034009921518671662384088, −2.40764666794402133491279627940, −1.29016524650164218258560436012, 0,
1.29016524650164218258560436012, 2.40764666794402133491279627940, 3.33572034009921518671662384088, 4.14803271888916308179694144893, 5.42214484723679331344623848665, 5.89515373704576002244865266302, 6.78110350529806184072952302084, 7.70213284734941395996731595369, 8.799088134994071790982740711543