L(s) = 1 | + 3.28·2-s − 3·3-s + 2.76·4-s − 6.75·5-s − 9.84·6-s − 17.1·8-s + 9·9-s − 22.1·10-s + 11·11-s − 8.30·12-s + 17.8·13-s + 20.2·15-s − 78.4·16-s + 122.·17-s + 29.5·18-s − 52.2·19-s − 18.7·20-s + 36.0·22-s − 128.·23-s + 51.4·24-s − 79.3·25-s + 58.4·26-s − 27·27-s − 216.·29-s + 66.4·30-s − 252.·31-s − 120.·32-s + ⋯ |
L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.346·4-s − 0.604·5-s − 0.669·6-s − 0.758·8-s + 0.333·9-s − 0.700·10-s + 0.301·11-s − 0.199·12-s + 0.380·13-s + 0.348·15-s − 1.22·16-s + 1.74·17-s + 0.386·18-s − 0.630·19-s − 0.209·20-s + 0.349·22-s − 1.16·23-s + 0.437·24-s − 0.635·25-s + 0.440·26-s − 0.192·27-s − 1.38·29-s + 0.404·30-s − 1.46·31-s − 0.664·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.054291736\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.054291736\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 - 3.28T + 8T^{2} \) |
| 5 | \( 1 + 6.75T + 125T^{2} \) |
| 13 | \( 1 - 17.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 469.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 124.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 449.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 356.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 253.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 812.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 183.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 902.T + 9.12e5T^{2} \) |
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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
−9.135181302083192069678922747461, −7.980934458643845708446323079788, −7.38736081556280120045001711448, −6.10960626660266995996380121456, −5.82303184198139954261755085425, −4.88074931177535107924110152629, −3.83045970827154279258847441858, −3.62606780508970286599844176311, −2.08244447670310889202783573521, −0.58514951374839736589511743491,
0.58514951374839736589511743491, 2.08244447670310889202783573521, 3.62606780508970286599844176311, 3.83045970827154279258847441858, 4.88074931177535107924110152629, 5.82303184198139954261755085425, 6.10960626660266995996380121456, 7.38736081556280120045001711448, 7.980934458643845708446323079788, 9.135181302083192069678922747461